title: Emotion (Carly Rae Jepsen album)

text: Following the sudden worldwide success of "Call Me Maybe" in 2012, Jepsen found that the song had become "[this] huge, ginormous thing that really overshadowed the rest of our project" and further singles issued from "Kiss" failed to gain traction. Jepsen viewed her predicament as an opportunity to retract from the spotlight and contemplate the direction of her next album. She met with her record label and management after The Summer Kiss Tour wrapped up in late 2013, stating: I want you to put your faith in me that I'll come back when it's ready,' and they did and I'm very lucky to have a team that wasn't about trying to mass produce things and was really more looking at the quality of it."


title: Emotion (Carly Rae Jepsen album)

text: Jepsen spent time regrouping; searching for a "detour" that came to be in the form of a Broadway role: "I thought, how amazing would it be to take a left turn, somehow, and still come back to this? [...] but 'left turn'—I didn't know what that meant." She was approached by the producers of "Rodgers + Hammerstein's Cinderella" to audition for the titular character, and was formally offered the part after auditioning in Los Angeles and passing callbacks in New York. Jepsen assumed the role for twelve weeks from February 2014 to June 2014, and during this time she decided to handle her own A&R. With help from guitarist Tavish Crowe, Jepsen began emailing artists she admired to see if they were interested in collaborating, including Tegan and Sara, Rostam Batmanglij of Vampire Weekend and Shellback.


title: Emotion (Carly Rae Jepsen album)

text: Under the pressure of matching "Call Me Maybes success, Jepsen recorded an entire indie-folk album during her run on "Cinderella" as an act of "rebellion". Jepsen felt that the demo project was developed for the wrong reasons–out of superficiality–and it was ultimately scrapped: "I think there is a natural rebellion when you have success in one area to completely rebel against that. I needed to get that out of my system, I think. I made really weird music." Jepsen's work on "Cinderella" provided valuable perspective in terms of being defined by a single attribute, particularly from her costar Fran Drescher: "Everyone still calls her 'the Nanny'. I realized you can't give in to other people's perception of you. Everyone's gonna be known for something.""["Emotion"] had to not be about trying to prove something. I feel like if you're writing music just to have a different identity in the public's eye, it's sort of the wrong motive. It's gotta be coming from a place of what you love and what you're passionate about."


title: Emotion (Carly Rae Jepsen album)

text: Due to her dissatisfaction with "Kiss"'s constrained timeframe of two months, Jepsen went into "Emotion" with the intent of crafting a project that was authentic and allowed her to approach different avenues. Whereas "Kiss" was developed in an acquiescent manner, as Jepsen was simply grateful for the opportunity, "Emotion" was spurred by her desire to take more creative control. Jepsen commented that much of the album is about "trying to get some power back" after exiting a relationship and subsequently settling down in New York. She sought to channel a "heart-on-your-sleeve sensibility" reminiscent of 1980s pop music after attending a Cyndi Lauper concert in Japan. These themes were further explored through "old-school" Prince and Madonna records on morning runs before "Cinderella" rehearsals: "What I loved was how potent some of those [1980s] lyrics were—how heart wrenching, how everyone's tea leaves are just right there on their sleeve. In music today, everything is a little more coy, but I wanted that romance and that fantasy, and I think that a lot of people [my] age do."


title: Emotion (Carly Rae Jepsen album)

text: An epiphany came to Jepsen after finishing "Emotion", the album's title track, where she found that a 1980s pop sound, combined with a more "alternative" production, was what she had been seeking. This fashioned the album's direction entirely–"Emotion" developed as a midway point between the "pure" pop she recorded in Los Angeles and the indie-folk effort that was scrapped in New York.


title: Emotion (Carly Rae Jepsen album)

text: Jepsen combed through Dev Hynes' discography after becoming infatuated with "Losing You" by Solange, to which she found his name listed in its production credits. Jepsen sought to collaborate, stating she was a fan, which Hynes hesitated to believe. He was eventually "won over" by Jepsen's demonstration of her vocal ability and work ethic, and credited her with genuine intentions of "[developing] a new aesthetic" versus pursuing ""Pitchfork"-approved artists" for the sake of indie credibility. The pair worked in a Chelsea studio between Jepsen's "Cinderella" performances. Hynes sent a demo of what would become "All That", where Jepsen wrote the bridge and produced the vocals herself. Ariel Rechtshaid was brought in for additional work on the song. In a similar affair, Jepsen's admiration of Sky Ferreira's work led her to Rechtshaid and the pair began meeting for coffee dates, figuring out a session date as he was in the midst of producing with Brandon Flowers. Rechtshaid further contributed to "Emotion" by aiding in the selection process of its track list and finalizing the production on its closer, "When I Needed You".


title: Emotion (Carly Rae Jepsen album)

text: Rostam Batmanglij was a prior fan of Jepsen's work when he reached out to her in the summer of 2014, after learning that she was in Los Angeles writing with various people. Jepsen was "over the moon" and the pair developed "Warm Blood" over the next year, the first verse recorded as Jepsen sat on the carpet of Batmanglij's home studio. Batmanglij announced the track's existence via Twitter on April 29, 2015.


title: Emotion (Carly Rae Jepsen album)

text: Jepsen spent a month in Sweden recording for the album, per her label's suggestion. These sessions materialized in its lead single "I Really Like You", written with Peter Svensson of the Cardigans, "Your Type" with Rami Yacoub and Carl Falk, and the Mattman & Robin-produced "Run Away with Me" which developed over two separate trips to the country. The duo flew to Los Angeles for one last session with Jepsen, finishing "Gimmie Love" in a day.


title: Emotion (Carly Rae Jepsen album)

text: Prior to starting full-fledged work on "Emotion", Jepsen had sketched several of its demos on her tour bus as she supported "Kiss" back in 2013. Of these, "Boy Problems" would eventually finalize its form with Sia and Greg Kurstin. Sia wrote the bridge to "Boy Problems", and contributed the outside cut "Making the Most of the Night", which Sia originally developed with the members of Haim. "I Didn't Just Come Here to Dance" existed as early as 2011 according to matching lyrics on a tweet authored by Jepsen. It would eventually be presented to Max Martin after the pair finished work on "Tonight I'm Getting Over You" in 2012; intending to cease their session at midnight, the song compelled Martin to stay for two extra hours. It was confirmed that she had worked with Josh Ramsay, Ryan Stewart, Benny Blanco and Stargate, though none of these producers appeared on the finalized project. A total of 250 songs were composed over the course of the album's development. Speaking to the "Herald", Jepsen stated that she worked on "Emotion" past deadline and finished right before its Japan release date on June 24, 2015.


title: Emotion (Carly Rae Jepsen album)

text: "Emotion" contains elements of synth-pop and new wave. "Consequence of Sound" summed the record up as a "more mature, sophisticated version of her ["Kiss"] hyperpop", Jepsen elaborating: "I wanted to kind of blur the lines of what [a pop album] needed to be." She found it a challenge to repurpose the album's 80s influences into a modern context without delving into "empty nostalgia", stating that the album is not strictly a "period piece", but is "tinted with shades of that era". The lyrics on "Emotion" "shade her old ebullience with darker, more complex feelings", and it is sonically grounded with "earthier textures" from 80s R&B–cleansing herself of the "Cinderella" performances during the record's production. Elsewhere, Jepsen explores funk and disco on "Boy Problems", and house music on "I Didn't Just Come Here to Dance".


title: Emotion (Carly Rae Jepsen album)

text: Jepsen singled "All That" out as most representative of the goals she sought to accomplish with "Emotion". Lyrically: "['All That'] holds a special place in my heart because it is so revealing: It's talking about the desire for intimacy with somebody. And I think with songs like 'Call Me Maybe', that can be quite light and a little bit more surfacey, it's fun to get a little deeper." Jepsen penned "Your Type" with Rami Yacoub and Carl Falk at four in the morning when she was "losing her mind": "They got me hooked for a week on those little fake cigarettes that taste of strawberries. You can hear it in my voice, I sound all gritty. It's because I was vaping for a week."


title: Emotion (Carly Rae Jepsen album)

text: "Warm Blood" was produced by Rostam Batmanglij and co-written with Tino Zolfo and Joe Cruz. It initially held the hook "warm love feels good" to which Batmanglij misheard as "warm blood", sticking as its main motif as he was drawn to its physical rather than abstract connotations. Jepsen explained: "The more we chipped away at it, we couldn't get away from how much better 'blood' felt and how realer it was. It's almost like that warm skin or that feeling of intimacy." The song has been noted for its experimental vocal effects, with Batmanglij scattering distorted vocal cuts throughout, a "creepy" quality that the pair immediately liked. Elsewhere, sections of "Warm Blood" were sung in a lower pitch; Jepsen was to re-record these parts as she came to the session with "shot vocals", however its "smokier" quality abandoned these plans. Brad Nelson of "The Guardian" compared Jepsen's performance to that of Ezra Koenig's "machine-produced flexibility" on the Vampire Weekend song, "Diane Young".


title: Emotion (Carly Rae Jepsen album)

text: In a session with Tavish Crowe, the closer "When I Needed You" was written to process a break up where Jepsen realizes the faults of her seemingly "perfect" relationship: "[...] but in order to stay in it, I would have to be quite a different person than who I naturally am [...] and that sacrifice didn't seem worth it in the end." Dan Nigro and Nate Campany composed the "happier-sounding" instrumental, to which Jepsen felt a sense of catharsis in concealing a "very serious" emotion. Ariel Rechsthaid reworked the chord progression in order to invoke a sense of "desperation", droning through it with a series of bell notes that made the composition sound "a little bit more somber". The "five-string, funk-R&B" bass line was played by Ethan Farmer, and the drum fill, "an 80s kind-of snare with a big reverb on it," was inspired by John Mellencamp's "Jack & Diane".


title: Emotion (Carly Rae Jepsen album)

text: "When I'm Alone", a song written and co-composed by Jepsen during sessions for "Emotion", was eventually purchased by SM Entertainment and given to K-pop girl group f(x) for their album "4 Walls". "Cut to the Feeling" and "Runaways", both written by Jepsen during sessions for "Emotion", were recorded for the soundtrack of the 2016 film "Ballerina". "Wildflowers", a song written during the sessions for "Emotion" which leaked online in 2016, was covered by Elle Fanning for the soundtrack of the film "Teen Spirit" (2018).


title: Emotion (Carly Rae Jepsen album)

text: "Eternal Summer" was a running contender for the album's title, in reference to Los Angeles being an "eternal summer in sunshine" where time perspective is lost. It originated from the song "Eternal Summer" which Jepsen developed for a scrapped indie-folk effort; the song was ultimately cut from "Emotion" as well. Per suggestion from her A&R, the song "Emotion" was retrofitted as the album's title as Jepsen was fond of its strength, both as a one-word title and its complexity as a concept. Jepsen further stated that the song "Emotion" itself encapsulated her feelings of clarity, as its writing process steered her in the direction of 80s emotional pop". Jepsen was "sold" on the title "Emotion" after she was sent its phonetic spelling, which is reflected in its stylization ("E·MO·TION").


title: Emotion (Carly Rae Jepsen album)

text: The album artwork features Jepsen sitting in a reserved position as she dons a technicolor sweater and black tights: "There were a few different pictures that had more of a decided facial expression, but I kind of liked the fact that I can't totally read what I was thinking in that picture. It could be many things, and this album, to me, was sort of a collection of many different emotions." The artwork's typography bears the dictionary entry of "Emotion" as a noun.


title: Emotion (Carly Rae Jepsen album)

text: Jepsen announced the title of the album on April 11, 2015, and released the cover artwork on April 15, 2015. The track listing was unveiled on June 2, 2015. The album's lead single, "I Really Like You", was released to the iTunes Store on March 2, 2015. The song reached the top 40 in Australia, Canada, Japan, and Netherlands, and has reached number 39 on the US "Billboard" Hot 100, number 3 in Ireland and on the UK Singles Chart, and number 1 on the Scottish Singles Chart. The music video was released on March 6, 2015, featuring Tom Hanks and Justin Bieber. The album's promotion was kicked off with a live performance of "Really Like You" at "Good Morning America" on March 5, 2015. "All That" was issued as the first promotional single on April 5, 2015. The following day, Jepsen performed the track at "Saturday Night Live".


title: Emotion (Carly Rae Jepsen album)

text: On May 1, 2015, Jepsen performed "Run Away with Me", "Emotion", "Your Type", "Black Heart" and "Gimmie Love" at a show in Beijing, China. The album's title-track was released on June 3, 2015. "Run Away with Me" and "Your Type" also premiered on a Spanish radio station prior to the album's release on June 22, 2015. "Run Away with Me" was released as the album's second single on July 17, 2015. The music video for "Run Away with Me" was released on July 17, 2015 and directed by Jepsen’s former boyfriend, David Kalani Larkins. The video was filmed in Tokyo, New York City, and Paris. The track was premiered on Hit FM in Spain on June 22, 2015. "Warm Blood" was released as the second promotional single on July 31, 2015. "Making the Most of the Night" was released as the third promotional single on August 7, 2015. "Your Type" was released as the fourth promotional single on August 14, 2015.


title: Emotion (Carly Rae Jepsen album)

text: On August 21, the release date of "Emotion" in the United States, Jepsen performed "Run Away with Me" on "Today". During her tour in South Africa in October 2015, she performed "Run Away With Me" on "Idols South Africa". "Your Type" was re-released as the third official single on November 9, 2015, in Europe. An official remix package was released on December 11, 2015, in Europe and Oceania and on December 18, 2015, in North America. Its music video, directed by Gia Coppola, premiered November 3, 2015, and follows Jepsen on a Cinderella-inspired story where her character fantasizes about becoming a pop star.


title: Emotion (Carly Rae Jepsen album)

text: A music video for "Boy Problems" premiered on April 8, 2016. It was directed by Petra Collins and stars Tavi Gevinson, Barbie Ferreira, Paloma Elsesser, among others.


title: Emotion (Carly Rae Jepsen album)

text: "Emotion" received positive reviews from music critics. At Metacritic, which assigns a normalized rating out of 100 to reviews from mainstream publications, the album received an average score of 77, based on 24 reviews. The album was praised for its "pop perfection" in catchiness, cohesion and production value. Sasha Geffen of "Consequence of Sound" wrote, "Few artists have taken a logarithmic hit like "Call Me Maybe" as a sign to push even further, to make something better, more human, and more electric. But Jepsen is the kind of singer who thrives on the stakes that unapologetic pop music offers." Annie Zaleski of "The A.V. Club" said, "If there's any justice, "Emotion" will propel her to superstardom—but even if it doesn't, she can at least rest easy knowing she made one of 2015's most interesting, effervescent records." Peter Tabakis of "Pretty Much Amazing" stated that ""Emotion" is so good, it's formed sky-high expectations out of thin air." "Slant Magazine"'s James Rainis writes, ""Emotion" is further proof that Jepsen is capable of translating broadly understood emotions and experiences into unshakable earworms."


title: Emotion (Carly Rae Jepsen album)

text: Opinion was divided over the album's lyrical content, which some reviewers have deemed as immature or bland. Corban Goble of "Pitchfork" commented, "It's unfair to deeply scrutinize lyrics on a pop record—the goal is to write smart, but skew broad—but "Emotion" fails to tell us who Jepsen is or wants to be." Her "absence of an identity" was further compared to her contemporaries for their image-conscious work. While Adam Downer of "cokemachineglow" opines that the album's "retro-pop bliss without angle or ego lends it a refreshingly timeless quality", further contexualizing it [in] "a year where pop stars fight for brand supremacy"; Alexis Petridis of "The Guardian" states that the issue isn't Jepsen being without an "outrageous, headline-grabbing persona" like Rihanna or Miley Cyrus but that "she doesn't do anything to stamp her identity on the songs [...] It's one problem that all the expensive names in the credits can't solve." Similarly, Jon Caramanica of "The New York Times" scrutinizes the heavy-lifting done by "Emotion"'s cast of collaborators: "Maybe Ms. Jepsen's choices merely reinforce the new centrist pop model of ’80s sleekness [...]; but why fall under the spell of someone else's cool when you can luxuriate in the stink of your own cheese?"


title: Emotion (Carly Rae Jepsen album)

text: In a more negative review, Billy Hamilton of "Under the Radar" critiques the poptimism narrative surrounding Jepsen and her perceived effort to appease "indie tastemakers", regarding her as "the pet project of a creative hipsterati that's determined to prove pop is cooler than you, or I, could possibly ever imagine," further expressing that "Carly Rae Jepsen and her production team try overly hard to be clever." In a similar conclusion, Evan Sawdey of "PopMatters" writes that ""Emotion" is still a very pleasing album if not just a shade overambitious, clearly trying too hard to make the same genius pop moments that "Kiss" churned them out with effortless flair."


title: Emotion (Carly Rae Jepsen album)

text: The album debuted at number 16 on the US "Billboard" 200, earning 16,153 album sales in its first week. By the end of 2015, "Emotion" had sold a total of 36,000 copies. In Canada, the album debuted at number 8 with 2,600 copies sold in its first week. In Japan, the album debuted at number 8 with 12,189 physical copies sold in its first week. On April 2, 2016, Jepsen revealed via Twitter that "Emotion" was certified gold by the Recording Industry Association of Japan, having sold over 100,000 copies there, these copies represent physical sales of the album only. The album entered the UK Albums Chart on September 25 at number 21 with sales of 4,150 copies in addition to streaming figures.


title: Emotion (Carly Rae Jepsen album)

text: "Emotion" is considered a crucial factor in Jepsen's "unlikely" career trajectory, following her stint on "Canadian Idol" and the ubiquity of "Call Me Maybe" to "cult idol". Marked as a transitional piece, publications commended Jepsen for cultivating her sound, which "[reestablished] herself as a pop star for grown-ups". Carrie Battan of the "New Yorker" posed that "Emotion" spared Jepsen from "falling to her death" and instead descended her to the bottom akin to a rising "mindie" artist, online buzz and "underground cred" in tow: "Jepsen, the woman behind one of the biggest songs of this century, now resembles someone whom she never had the opportunity to become at the beginning: an indie darling."


title: Emotion (Carly Rae Jepsen album)

text: "Emotion" was labelled a "commercial flop" as its promotional cycle waned. James Rettig of "Stereogum" writes, "The lead-up to "Emotion" played out like a lesson in what "not" to do with a pop singer sitting precariously on the edge between cultural ubiquity and cult following." Some blamed its promotional roll-out, with a Japanese release arriving two months ahead and therefore susceptible to leaks. Elsewhere, others focused on creative choices–Rettig criticized "I Really Like You" as lead single, the "most damaging misstep" that hindered Jepsen's ability to showcase her artistic growth. "FasterLouder"'s Jules LeFevre noted that Jepsen's decision against promoting any archetypal identity rendered herself "indistinct" in the "crowded pop landscape".


title: Emotion (Carly Rae Jepsen album)

text: The record grew to become a cult hit with Jepsen labelled as an "underdog" as it spread by word-of-mouth. Whereas music critics were "compelled by the narrative of a one-hit wonder trying to rebuild herself", as quipped by Battan, Caitlin White of Uproxx writes, "I think "Emotion" wouldn't be as meaningful if we had to share it with capitalism's steely machinery; its commercial failure is part of what makes it continue to feel intimate, ours." "Emotion" is noted for garnering Jepsen a large LGBTQ audience, Brandon Tensley of "Pacific Standard" opining that her music "taps into a shared queer history of escape, possibility, and disappointment", likening her to Kylie Minogue.


title: Emotion (Carly Rae Jepsen album)

text: NPR called "Emotion" a "modern touchstone", by which they compared the Aces and Paramore's "After Laughter" (2017) to. In developing her EP "Now That the Light Is Fading", Maggie Rogers resolved to make pop music after being inspired by "Emotion". Jay Som counts Jepsen as a notable influence on her debut album "Everybody Works": "I felt very assertive with ["Everybody Works"] because I was also listening to her music. I liked how energetic and youthful the spirit is—and it's just so not ashamed to be this pop record." A tribute album was released digitally by Something Merry on November 21, 2018. The album is a track-by-track cover of "Emotion", including covers made by Wild Pink, Cheer Up, Future Teens, Gabe Goodman, Good Looking Friends, Kiki Maddog, Lilith, Mandancing, oldsoul, Photocomfort, Pushflowers, the Aux, the Superweaks, Tuft.


title: Roger Mortimer, 1st Earl of March

text: Roger Mortimer, 3rd Baron Mortimer of Wigmore, 1st Earl of March (25 April 1287 – 29 November 1330), was an English nobleman and powerful Marcher Lord who gained many estates in the Welsh Marches and Ireland following his advantageous marriage to the wealthy heiress Joan de Geneville, 2nd Baroness Geneville.


title: Roger Mortimer, 1st Earl of March

text: In November 1316, he was appointed Lord Lieutenant of Ireland. He was imprisoned in the Tower of London in 1322 for having led the Marcher lords in a revolt against King Edward II in what became known as the Despenser War.


title: Roger Mortimer, 1st Earl of March

text: He later escaped to France, where he was joined by Edward's queen consort Isabella, where they may have begun an affair. After he and Isabella led a successful invasion and rebellion, Edward was deposed; Mortimer allegedly arranged his murder at Berkeley Castle.


title: Roger Mortimer, 1st Earl of March

text: For three years, Mortimer was "de facto" ruler of England before being himself overthrown by Edward's eldest son, Edward III. Accused of assuming royal power and other crimes, Mortimer was executed by hanging at Tyburn.


title: Roger Mortimer, 1st Earl of March

text: Mortimer, grandson of Roger Mortimer, 1st Baron Mortimer of Wigmore, and Maud de Braose, was born at Wigmore Castle, Herefordshire, England, the firstborn of Marcher Lord Edmund Mortimer, 2nd Baron Mortimer of Wigmore, and Margaret de Fiennes. He was born on 25 April 1287, the Feast of Saint Mark, a day of bad omen. He shared this birthday with King Edward II, which would be relevant later in life.


title: Roger Mortimer, 1st Earl of March

text: Edmund Mortimer was a second son, intended for minor orders and a clerical career, but on the sudden death of his elder brother Ralph, Edmund was recalled from Oxford University and installed as heir. According to his biographer Ian Mortimer, Mortimer was possibly sent as a boy away from home to be fostered in the household of his formidable uncle, Roger Mortimer de Chirk. It was this uncle who had carried the severed head of Llywelyn ap Gruffudd of Wales to King Edward I in 1282.


title: Roger Mortimer, 1st Earl of March

text: Mortimer attended the Coronation of Edward II on 25 February 1308 and carried a table bearing the royal robes in the ceremony's procession.


title: Roger Mortimer, 1st Earl of March

text: Like many noble children of his time, Mortimer was betrothed at a young age, to Joan de Geneville (born 1286), the daughter of Sir Peter de Geneville, of Trim Castle and Ludlow. They were married on 20 September 1301 when he was aged fourteen. Their first child was born in 1302.


title: Roger Mortimer, 1st Earl of March

text: Through his marriage, Mortimer not only acquired numerous possessions in the Welsh Marches, including the important Ludlow Castle, which became the chief stronghold of the Mortimers, but also extensive estates and influence in Ireland. However, Joan de Geneville was not an "heiress" at the time of her marriage. Her grandfather Geoffrey de Geneville, at the age of eighty in 1308, conveyed most, but not all, of his Irish lordships to Mortimer, and then retired: he finally died in 1314, with Joan succeeding as "suo jure" 2nd Baroness Geneville. During his lifetime Geoffrey also conveyed much of the remainder of his legacy, such as Kenlys, to his younger son Simon de Geneville, who had meanwhile become Baron of Culmullin through marriage to Joanna FitzLeon. Mortimer, therefore, succeeded to the eastern part of the Lordship of Meath, centred on Trim and its stronghold of Trim Castle. He did not succeed, however, to the Lordship of Fingal.


title: Roger Mortimer, 1st Earl of March

text: Mortimer's childhood came to an abrupt end when his father was mortally wounded in a skirmish near Builth in July 1304. Since Mortimer was underage at the death of his father, he was placed by King Edward I under the guardianship of Piers Gaveston, 1st Earl of Cornwall. However, on 22 May 1306, in a lavish ceremony in Westminster Abbey with two hundred and fifty-nine others, he was knighted by Edward and granted livery of his full inheritance.


title: Roger Mortimer, 1st Earl of March

text: His adult life began in earnest in 1308, when he went to Ireland in person to enforce his authority. This brought him into conflict with the de Lacys, who turned for support to Edward Bruce, brother of Robert Bruce, King of Scots. Mortimer was appointed Lord Lieutenant of Ireland by Edward II on 23 November 1316. Shortly afterwards, at the head of a large army, he drove Bruce to Carrickfergus and the de Lacys into Connaught, wreaking vengeance on their adherents whenever they were to be found. He returned to England and Wales in 1318 and was then occupied for some years with baronial disputes on the Welsh border.


title: Roger Mortimer, 1st Earl of March

text: Mortimer became disaffected with his king and joined the growing opposition to Edward II and the Despensers. After the younger Despenser was granted lands belonging to him, he and the Marchers began conducting devastating raids against Despenser property in Wales. He supported Humphrey de Bohun, 4th Earl of Hereford, in refusing to obey the king's summons to appear before him in 1321 as long as ""the younger Despencer was in the King's train."" Mortimer led a march against London, his men wearing the Mortimer uniform which was green with a yellow sleeve. He was prevented from entering the capital, although his forces put it under siege. These acts of insurrection compelled the Lords Ordainers led by Thomas, 2nd Earl of Lancaster, to order the king to banish the Despensers in August. When the king led a successful expedition in October against Margaret de Clare, Baroness Badlesmere, after she had refused Queen Isabella admittance to Leeds Castle, he used his victory and new popularity among the moderate lords and the people to summon the Despensers back to England. Mortimer, in company with other Marcher Lords, led a rebellion against Edward, which is known as the Despenser War.


title: Roger Mortimer, 1st Earl of March

text: In January 1322, Mortimer attacked and burnt Bridgnorth but, being heavily outnumbered, was forced to surrender to the king at Shrewsbury. Mortimer joined Lancaster at the Battle of Boroughbridge in March 1322 and warrants for his arrest were issued in July.


title: Roger Mortimer, 1st Earl of March

text: A death sentence was passed upon Mortimer but this was commuted to life imprisonment and he was consigned to the Tower of London. In August 1323 Mortimer, aided by Gerald de Alspaye, the sub-lieutenant or valet of the Tower's Constable, drugged the warders during a feast, allowing Mortimer to escape. He attempted to capture Windsor and Wallingford Castles to free imprisoned Contrariants. Mortimer eventually fled to France, pursued by warrants for his capture dead or alive.


title: Roger Mortimer, 1st Earl of March

text: In the following year Queen Isabella, anxious to escape from her husband, obtained his consent to her going to France to use her influence with her brother, King Charles IV, in favour of peace. At the French court, the queen found Mortimer, who became her lover soon afterwards. At his instigation, she refused to return to England so long as the Despensers retained power as the king's favourites.


title: Roger Mortimer, 1st Earl of March

text: Historians have speculated as to the date at which Mortimer and Isabella actually became lovers. The modern view is that the affair began while both were still in England, and that after a disagreement, Isabella abandoned Mortimer to his fate in the Tower. His subsequent escape became one of medieval England's most colourful episodes. However, almost certainly Isabella risked everything by chancing Mortimer's companionship and emotional support when they first met again at Paris four years later (Christmas 1325). King Charles IV's protection of Isabella at the French court from Despenser's would-be assassins played a large part in developing the relationship.


title: Roger Mortimer, 1st Earl of March

text: In 1326, Mortimer moved as Prince Edward's guardian to Hainault, but only after a furious dispute with the queen, demanding she remain in France.


title: Roger Mortimer, 1st Earl of March

text: Isabella retired to raise troops in her County of Ponthieu; Mortimer arranged the invasion fleet supplied by the Hainaulters and an army supplied by his supporters back in England, who had been sending him aid and advice since at least March 1326.


title: Roger Mortimer, 1st Earl of March

text: The scandal of Isabella's relations with Mortimer compelled them both to withdraw from the French court to Flanders, where they obtained assistance for an invasion of England from Count William of Hainaut, although Isabella did not arrive from Ponthieu until the fleet was due to sail. Landing in the River Orwell on 24 September 1326, they were accompanied by Prince Edward and Henry, Earl of Lancaster. London rose in support of the queen, and Edward took flight to the west, pursued by Mortimer and Isabella. After wandering helplessly for some weeks in Wales, the king was taken prisoner on 16 November, and was compelled to abdicate in favour of his son. Though the latter was crowned as Edward III of England on 1 February 1327, the country was ruled by Mortimer and Isabella. On 21 September that same year, Edward II died in captivity. The suspicious death of Edward II has been the subject of many conspiracy theories, including that Mortimer's men killed him, but none has been proven.


title: Roger Mortimer, 1st Earl of March

text: Following the removal of the Despensers, Mortimer set to work in restoring the status of his supporters, primarily in the Marches, and hundreds of pardons and restorations of property were made in the first year of the new king's reign. Rich estates and offices of profit and power were heaped on Mortimer. He was made constable of Wallingford Castle and in September 1328 he was created Earl of March. However, although in military terms he was far more competent than the Despensers, his ambition was troubling to all. His own son Geoffrey, the only one to survive into old age, mocked him as "the king of folly" deriding his ambitious extravagance of "rich clothes ot of manner resoun, both of shaping and wearing". During his short time as ruler of England he took over the lordships of Denbigh, Oswestry and Clun (the first of which belonged to Despenser, the latter two had been the Earl of Arundel's). He was also granted the "marcher" lordship of Montgomery by the queen. During the War of Saint Sardos, the Regent and his queen spent over £60,000 bankrupting the Treasury, even after the proscriptions of Arundel and the Despensers. The Lancastrian opposition were incensed by this casual display of irresponsible government.


title: Roger Mortimer, 1st Earl of March

text: The jealousy and anger of many nobles were aroused by Mortimer's use of power, which in many ways was tenuous. In 1328 Simon de Mepham, reportedly a Lancastrian at court, was elected Archbishop of Canterbury without controversy. However, the feuding would not stop. The day Parliament opened on 15 October Thomas of Lancaster's nemesis, Sir Robert Holland was murdered by highway robbers. Whereupon March swore on Mepham's cross that he knew nothing of it. Nonetheless, the King decreed an indictment; he would be judged at law against the standards of the Magna Carta. With Parliament adjourned on 31 October, he was able to slip away to his estates on the Marches. The two earls’ lethal enmity, and enforced absence from the King's presence, rendered their motives almost equally suspect to rowdy Londoners. The young king would have to raise an army of archers if he was to defend his throne from a northern rebellion controlled by Lancaster. In charge of the army, Lancaster blamed Mortimer and his queen for the debacle, and the highly contentious Treaty of Edinburgh–Northampton with the Scots.


title: Roger Mortimer, 1st Earl of March

text: Henry, Earl of Lancaster, one of the principals behind Edward II's deposition, tried to overthrow Mortimer, but the action was ineffective as the young king passively stood by. Then, in March 1330, Mortimer ordered the execution of Edmund, Earl of Kent, the half-brother of Edward II. After this execution Henry Lancaster prevailed upon the young king, Edward III, to assert his independence. In October 1330, a Parliament was summoned to Nottingham, just days before Edward's eighteenth birthday, and Mortimer and Isabella were seized by Edward and his companions from inside Nottingham Castle. In spite of Isabella's entreaty to her son, "Fair son, have pity on the gentle Mortimer", Mortimer was conveyed to the Tower. Accused of assuming royal power and of various other high misdemeanours, he was condemned without trial and hanged at Tyburn on 29 November 1330, his vast estates forfeited to the crown. His body hung at the gallows for two days and nights in full view of the populace. Mortimer's widow Joan received a pardon in 1336 and survived until 1356. She was buried beside Mortimer at Wigmore, but the site was later destroyed.


title: Roger Mortimer, 1st Earl of March

text: In 2002, the actor John Challis, the owner of the remaining buildings of Wigmore Abbey, invited the BBC programme "House Detectives at Large" to investigate his property. During the investigation, a document was discovered in which Mortimer's widow Joan petitioned Edward III for the return of her husband's body so she could bury it at Wigmore Abbey. Mortimer's lover Isabella had buried his body at Greyfriars in Coventry following his hanging. Edward III replied, "Let his body rest in peace". The king later relented, and Mortimer's body was transferred to Wigmore Abbey, where Joan was later buried beside him.


title: Roger Mortimer, 1st Earl of March

text: The marriages of Mortimer's children (three sons and eight daughters) cemented Mortimer's strengths in the West.


title: Roger Mortimer, 1st Earl of March

text: Through his son Sir Edmund Mortimer, he is an ancestor of the last Plantagenet monarchs of England from kings Edward IV to Richard III. By Edward IV's daughter, Elizabeth of York, the Earl of March is an ancestor to King Henry VIII of England and King James V of Scotland, and therefore to all subsequent Scottish, English and British monarchs.


title: Roger Mortimer, 1st Earl of March

text: Mortimer appears in Christopher Marlowe's play "Edward II" (c. 1592), the Restoration tragedy "Edward III" (1690), as well as Bertolt Brecht's "The Life of Edward II of England" (1923). In Derek Jarman's film "Edward II" (1991), based on Marlowe's play, he is portrayed by Nigel Terry.


title: Roger Mortimer, 1st Earl of March

text: Mortimer is also a character in "Les Rois maudits" ("The Accursed Kings"), a series of French historical novels by Maurice Druon. He was portrayed by Claude Giraud in the 1972 French miniseries adaptation of the series, and by Bruno Todeschini in the 2005 adaptation. Mortimer is also briefly mentioned in the 2001 film "A Knight's Tale" as a contestant in a joust.


title: Edward Bluemel

text: Edward Bluemel is an English actor best known for his portrayal of Marcus Whitmore in Sky One's adaptation of "A Discovery of Witches" (2018), based on the book of the same name by Deborah Harkness. Bluemel was also featured in "The Halcyon "(2017), "The Commuter" (2018), "Sex Education" (2019) and "Killing Eve" (2019).


title: Edward Bluemel

text: He attended Taunton School until 2011, and is an alumnus of the Royal Welsh College of Music & Drama with a BA in Acting.


title: Mealybug

text: Mealybugs are insects in the family Pseudococcidae, unarmored scale insects found in moist, warm habitats. Many species are considered pests as they feed on plant juices of greenhouse plants, house plants and subtropical trees and also act as a vector for several plant diseases. Some ants live in symbiotic relationships with them, protecting them from predators and feeding off the honeydew which they excrete.


title: Mealybug

text: Mealybugs are sexually dimorphic: females appear as nymphs, exhibiting reduced morphology, and lack wings, although unlike many female scale insects, they often retain legs and can move. Males are smaller, gnat-like and have wings. Since mealybugs (as well as all other Hemiptera) are hemimetabolous insects, they do not undergo complete metamorphosis in the true sense of the word. However, male mealybugs do exhibit a radical change during their life cycle, changing from wingless, ovoid nymphs to wasp-like flying adults.


title: Mealybug

text: Mealybug females feed on plant sap, normally in roots or other crevices, and in a few cases the bottoms of stored fruit. They attach themselves to the plant and secrete a powdery wax layer (hence the name "mealy" bug) used for protection while they suck the plant juices. In Asia, mango mealybug is considered a major menace for the mango crop. The males on the other hand are short-lived as they do not feed at all as adults and only live to fertilize the females. Male citrus mealy bugs fly to the females and resemble fluffy gnats.


title: Mealybug

text: Some species of mealybug lay their eggs in the same waxy layer used for protection in quantities of 50–100; other species are born directly from the female.


title: Mealybug

text: The most serious pests are mealybugs that feed on citrus; other species damage sugarcane, grapes, pineapple (Jahn et al. 2003), coffee trees, cassava, ferns, cacti, gardenias, papaya, mulberry, sunflower and orchids. Mealybugs only tend to be serious pests in the presence of ants because the ants protect them from predators and parasites. Mealybugs are also a vector of viruses in grapevines, spreading grapevine leafroll and grapevine red blotch viruses. Mealybugs also infest some species of carnivorous plant such as "Sarracenia" (pitcher plants); in such cases it is difficult to eradicate them without repeated applications of insecticide such as diazinon. Small infestations may not inflict significant damage. In larger amounts though, they can induce leaf drop. In recent years, some of the mealybug species have become invasive pests in localities posing a great problem to the new agro-ecosystems. In India, "Withania somnifera" plant have been reported as a new reservoir host for an invasive mealybug species "Phenacoccus solenopsis".


title: Mealybug

text: Some mealybugs of the "Hypogeococcus" are used as biological pest controls of invasive cacti in South Africa, including "Harrisia balansae", "H. martinii", and "Opuntia cespitosa".


title: Mealybug

text: Fossil specimens of genus "Acropyga" ants have been recovered from the Burdigalian stage Dominican amber deposits and several individuals are preserved carrying the extinct mealybug genus "Electromyrmococcus". These fossils represent the oldest record of the symbiosis between mealybugs and "Acropyga" species ants.


title: Mealybug

text: Some gardeners use green lacewing (Chrysopidae) larvae to control mealybug infestations, as the larval lacewings are voracious predators of aphids and other small insects.


title: Mealybug

text: Mealybugs have a highly sophisticated metabolism that involves not one but two bacterial endosymbionts, one inside the other. The endosymbionts make essential amino acids that the mealybug is now unable to acquire directly from its diet. Genetically, mealybugs rely on a "mosaic" of metabolic pathways in which proteins are transported across membranes between what were once independent organisms.


title: Ugandan Bush War

text: The Ugandan Bush War, also known as the Luwero War, the Ugandan Civil War or the Resistance War, was a civil war fought in Uganda by the official Ugandan government and its armed wing, the Uganda National Liberation Army (UNLA), against a number of rebel groups, most importantly the National Resistance Army (NRA), from 1980 to 1986.


title: Ugandan Bush War

text: The unpopular President Milton Obote was overthrown in a coup d'état in 1971 by General Idi Amin, who established a military dictatorship. Amin was overthrown in 1979 following the Uganda-Tanzania War, but his loyalists started the Bush War by launching an insurgency in the West Nile region in 1980. Subsequent elections saw Obote return to power in a UNLA-ruled government. Several opposition groups claimed the elections were rigged, and united as the NRA under the leadership of Yoweri Museveni to start an armed uprising against Obote's government on 6 February 1981. Obote was overthrown and replaced as president by his general Tito Okello in 1985 during the closing months of the conflict. Okello formed a coalition government consisting of his followers and several armed opposition groups, which agreed to a peace deal. In contrast, the NRA refused to compromise with the government, and conquered much of western and southern Uganda in a number of offensives from August to December 1985.


title: Ugandan Bush War

text: The NRA captured Kampala, Uganda's capital, in January 1986. It subsequently established a new government with Museveni as president, while the UNLA fully disintegrated in March 1986. Obote and Okello went into exile. Despite the nominal end of the civil war, numerous anti-NRA rebel factions and militias remained active, and would continue to fight Museveni's government in the next decades.


title: Ugandan Bush War

text: In 1971, the President of Uganda Milton Obote was overthrown in a coup d'état by parts of the Uganda Army which put Idi Amin in power. Obote had been president since Uganda's independence from the United Kingdom in 1962, and his regime saw a general decline in living standards in the country, with growing corruption, factional violence, and persecution of ethnic groups. Obote's increasing unpopularity led him to believe rivals were beginning to plot against him, particularly Amin and arranged a purge to occur while he was outside of the country. As Amin was popular in sections of the military, his loyalists responded by acting first and overthrowing the government, forcing Obote into exile in Tanzania. Despite initial popularity, Amin quickly turned to despotism and established a military dictatorship which accelerated the decline of Obote's regime, destroying the country's economy and political system.


title: Ugandan Bush War

text: As time went on, Amin's regime was increasingly destabilized by factionalism and economic decline, while opposition groups as well as dissatisfied elements of the Uganda Army repeatedly attempted to organize uprisings or to overthrow his regime by other means. Several opposition factions, including Obote's loyalists, were supported by Tanzania under President Julius Nyrere. In 1978, parts of the Uganda Army launched an invasion of Tanzania under unclear circumstances, resulting in open war with the neighboring country. Tanzania halted the assault, mobilised anti-Amin opposition groups, and launched a counter-offensive. Amin's forces and his Libyan allies were defeated by Tanzanian troops and the Uganda National Liberation Front (UNLF), a political coalition formed by exiled anti-Amin Ugandans under the leadership of Obote, whose armed wing was known as Uganda National Liberation Army (UNLA). Amin was overthrown during the fall of Kampala and then fled the country, and UNLF was installed by Tanzania to replace him. The unstable UNLF government ruled the country provisionally from April 1979 until December 1980. Meanwhile, the ousted Amin loyalists who had fled into Zaire and Sudan reorganised, and prepared to renew war in order to regain control of Uganda.


title: Ugandan Bush War

text: Meanwhile, Uganda's northeast was destabilized by large-scale banditry and communal violence. Karamojong groups, Uganda Army remnants, and foreign raiders used the political instability to raid cattle and other foodstock. These events caused a famine in Karamojong Province which killed 50,000 out of the 360,000 inhabintants of the northeastern highlands.


title: Ugandan Bush War

text: The first group to initiate hostilities were the Amin loyalists who launched a rebellion against the UNLF government in the autumn of 1980. Their 7,100-strong force never adopted an official name, but is generally called "Uganda Army" as it consisted for the most part of old troops of Amin's Uganda Army (it was also known as "West Front" or "Western Nile Front"). The rebels were not truly unified but split into several bands that were loyal to numerous officers who had previously served under Amin such as Emilio Mondo, Isaac Lumago, Isaac Maliyamungu, Elly Hassan, Christopher Mawadri, and Moses Ali. Amin arranged for the group to receive money from Saudi Arabia in preparation for a large-scale attack across the border against the West Nile sub-region. On 6 October, one week before the offensive was to commence, about 500 rebels crossed the border and attacked the town of Koboko. The 200-strong UNLA garrison was on parade at the time and was unarmed; the rebels massacred the soldiers. Word of the attack spread to other UNLA garrisons in West Nile, who quickly fled to the Nile River, leaving the Uganda Army's advance unopposed. They were welcomed by the local population, which had tense relations with the UNLA. As the rebels knew that they could not hold the captured territory against a full UNLA counter-offensive, they mostly retreated back into Sudan after a few days with a large amount of loot. The UNLA began its counterattack on 12 October accompanied by Tanzanian forces. The only significant resistance they encountered was in Bondo, where six Tanzanians were killed. The UNLA forces, considering the local population hostile, engaged in a campaign of destruction and looting across the West Nile, as Tanzanian officers tried in vain to restrain them. They leveled the town of Arua, killed over 1,000 civilians, and provoked the flight of over 250,000 refugees to Sudan and Zaire. The brutality of the UNLA inspired further unrest, as peasants and ex-soldiers took up arms to defend their lands from the government forces.


title: Ugandan Bush War

text: The Uganda Army launched its next offensive just before the Ugandan national elections in December 1980. In one of their most daring actions, the rebels ambushed Obote as he was touring the West Nile region. They almost killed him and Tito Okello, a high-ranking UNLA commander. This time, the Uganda Army also held the areas it captured in West Nile, and set up a parallel government after retaking Koboko. After about one month of combat, the insurgents had captured most of West Nile, leaving only some towns under UNLA control. However, many rebels focused more on looting the area and taking the plunder back to Zaire and Sudan than on fighting the UNLA.


title: Ugandan Bush War

text: The West Nile rebellion was weakened by internal divisions as parts of the Uganda Army remained loyal to Idi Amin, whereas others wanted to distance themselves from the unpopular old dictator. The latter part of the insurgent army split off, forming the "Uganda National Rescue Front" (UNRF) under Moses Ali, whereas the remaining Amin loyalists were still called the "Ugandan Army" until becoming known as "Former Uganda National Army" (FUNA). The West Nile rebels soon started to fight each other.


title: Ugandan Bush War

text: Furthermore, southwestern Uganda experienced a resurgence of the Rwenzururu movement which wanted self-determination for the Konjo and Amba peoples. The movement had been largely dormant since the 1960s, but managed to take control of weapon stockpiles that had been left unguarded when Amin's government collapsed in 1979. They thus resumed their insurgency, and the security situation in the mountainous border areas of the southwest quickly deteriorated in 1980.


title: Ugandan Bush War

text: Meanwhile, the UNLF government experienced its own divisions. As the UNLA was being transformed from a loose alliance of various anti-Amin insurgent groups into a regular army, the different political factions attempted to ensure that their own loyalists would be present and dominant in the new military. Obote outmaneuvered his rivals, most importantly Yoweri Museveni, and made his 5,000-strong Kikosi Maalum group the core of the UNLA. In contrast, just 4,000 out of Museveni's 9,000 FRONASA fighters were allowed to join the new army, and these were distributed across several units. Furthermore, FRONASA was forced to give up its own weaponry. At the same time, the UNLA was rapidly expanded; most of the new recruits came from ethnic groups that supported Obote. As result, power shifted to pro-Obote elements in the government and the army.


title: Ugandan Bush War

text: The elections of December 1980 were officially won by Milton Obote's Uganda People's Congress, effectively making him president of Uganda again. However, the results were strongly disputed by other candidates, resulting in increasing strife. Several political factions claimed electoral fraud, and believed themselves to be proven correct when Obote immediately launched a campaign of political repression. As the UNLA was dominated by pro-Obote forces, a coup was impossible, so the opposition instead launched armed rebellions against Obote's government: Museveni's followers created the Popular Resistance Army (PRA), Yusuf Lule formed the Uganda Freedom Fighters (UFF), Andrew Kayira armed his Uganda Freedom Movement (UFM), and the Communist "Gang of Four" organized an armed group known as the Uganda National Liberation Front – Anti-Dictatorship (UNLF-AD).


title: Ugandan Bush War

text: On 6 February 1981, hostilities began in the south with a PRA attack on the Kabamba Military Barracks in the central Mubende District. The attack aimed at capturing weaponry; although the operation failed to capture the armoury, Museveni's group of fighters managed to take a few guns and several vehicles before retreating. The PRA had more success during a series of attacks on police stations during the next days. Regardless, the small rebel group also came under pressure from UNLA and TPDF counter-insurgency operations, and still lacked a proper base. Museveni was familiar with guerrilla warfare, having fought with the Mozambican Liberation Front in Mozambique. He also had commanded his own Front for National Salvation to fight the Amin regime and had continued to campaign in rural areas hostile to Obote's government, especially central and western Buganda and in the western regions of Ankole and Bunyoro. As a result, he employed the tactics of a people's war to keep his small force active. In this regard, the PRA succeeded, as it won over many locals in the area around Kampala who considered Obote's government a regime which solely served the northerners. Without extensive support by sympathetic civilians during their early insurgency, Museveni's troops would have been easily crushed in 1981.


title: Ugandan Bush War

text: The PRA also enjoyed very limited foreign support. Some believed that Museveni was aided by his old Mozambican allies, resulting in tensions between Obote's government and Mozambique. Most attacks by Museveni's force involved small mobile units called "coys" under the command of Fred Rwigyema, and Museveni's brother, Salim Saleh, with "A" Coy led by Steven Kashaka, "B" Coy by Joram Mugume, and "C" Coy by Pecos Kuteesa. There were three small zonal forces: the Lutta Unit operating in Kapeeka, the Kabalega Unit operating near Kiwoko, and the Nkrumah Unit operating in the areas of Ssingo. Many of the early members of the PRA like Rwigyema and Paul Kagame were actually Rwandan refugees living in Uganda. They later organized the Rwandan Patriotic Front.


title: Ugandan Bush War

text: In contrast to Museveni's forces and the West Nile rebels, Andrew Kayira's UFM mostly consisted of relatively well-trained ex-soldiers and was focused on high-profile urban operations. The group hoped to destabilize Obote's government through direct attacks, a strategy which "doomed it to fail from the beginning" according to historians Tom Cooper and Adrien Fontanellaz. The UFM was not strong enough to challenge the UNLA head-on, suffered from leadership rivalries, lacked a firm organization, and was prone to being infiltrated by pro-government spies.


title: Ugandan Bush War

text: As the war escalated, foreign support became vital for the survival of Obote's government. The Tanzanians initially helped to defend his regime and kept some order through the presence of about 10,000 Tanzania People's Defence Force soldiers and 1,000 policemen. Nevertheless, the unsustainable costs of these troops led Tanzanian President Julius Nyerere to gradually withdraw most of his forces from Uganda. By June of 1981, just 800 to 1,000 Tanzanian advisors remained in the country. These advisors remained of crucial importance for the UNLA, but the Tanzanian withdrawal greatly weakened Obote's position. To compensate, he tried to enlist further foreign aid: He hired a British private military company, and convinced the Commonwealth of Nations as well as the United States to send small teams of security advisors.


title: Ugandan Bush War

text: One of Obote's most important allies was North Korea. The Ugandan President visited the country in late 1981 and signed a cooperation agreement which included military support for his regime. At least 30 North Korean officers were subsequently sent to Gulu in northern Uganda, where they trained UNLA soldiers and repaired military equipment. By 1984, the number had risen to about 50 North Koreans who acted as security, intelligence, and military advisors. Museveni claimed that over 700 North Koreans were ultimately used by the UNLA; Obote maintained that only about 170 were present in Uganda. According to one study, the North Korean officers actively participated in and even led counter-insurgency operations for Obote. A Central Intelligence Agency report stated, however, that the North Koreans refused to actually venture to the frontlines. The Ugandan military also sent some officers and non-commissioned officer to North Korea for advanced trainings. Obote's government also organized various paramilitary groups to assist the UNLA. The "People's Militia" consisted of Langi, Acholi and Teso tribesmen, and was mostly loyal to UNLA chief of staff David Oyite-Ojok. It became increasingly powerful, and garnered a reputation as a fierce and brutal force. In addition, there was the "National Youth Army" (NYA), various tribal militias, and the UPC youth paramilitaries.


title: Ugandan Bush War

text: Meanwhile, the conflict in the south became more serious. Another rebel group emerged, the so-called "Uganda Liberation Movement" which threatened to kidnap and kill United Nations personnel, as the latter was supporting Obote's attempts at restabilizing Uganda. The threats worked, and the U.N. stopped its training programme for the Ugandan police. The PRA also continued its hit-and-run operations with mixed successes: It overran a UNLA outpost at Kakiri on 5 April 1981, and captured important weaponry, but had to hastily retreat when a TPDF unit responded to the attack. The Tanzanian forces consequently conducted a counter-insurgency sweep, catching a PRA column under Elly Tumwine off-guard and recapturing some of the captured guns. Despite this, the PRA successfully recruited more volunteers, growing to about 200 fighters by early May. In the next month, Museveni travelled to Nairobi where he met with Lule; the two agreed to unite the PRA and UFF into a unified opposition group. The umbrella organization was dubbed the "National Resistance Movement" (NRM) and its armed wing was termed the "National Resistance Army" (NRA). Lule was appointed overall NRM Chairman, while Museveni became Vice-Chairman of the National Resistance Council and Chairman of the High Command of the NRA. The merger profited both sides: The UFF was extremely weak, and Lule finally gained an actual armed following, while Museveni was provided with important legitimacy, as Lule remained much respected among Uganda's southern population. This was especially important because a strategically important region near Kampala, known as Luwero Triangle, was mostly inhabited by Baganda. So far, the PRA had mostly consisted of non-Baganda, but Lule provided the newly formed NRA with backing from the Baganda, allowing Museveni to expand his envisioned "people's war". The Luwero Triangle consequently became the NRA's main operations area, although the group's center of recruitment remained Ankole in the west. Soon after the merger, Museveni implemented a strict code of conduct for fighters, allowing the NRA to remain highly disciplined and focused despite gradually growing in numbers and absorbing other insurgent factions such as the remnants of the "Gang of Four". By December 1981, the NRA had grown to about 900 militants. Libya's leader Muammar Gaddafi also opted to provide support to the NRA, although it was made up of forces which had overthrown his old ally Amin. Gaddafi believed that Libya could gain greater influence in central Africa through the NRA than it had previously with Amin. The Libyan support remained very limited, however, with about 800 rifles, a few machine guns and land mines being provided to the NRA in August 1981. Gaddafi demanded that the NRA merge with the UFM and UNRF to receive more substantial support, but the rebels remained rivals and refused to unite. Libya consequently cancelled its support to the NRA in 1982.


title: Ugandan Bush War

text: While the rebellion in the south grew in intensity, most of the West Nile region remained under rebel control. A local administration began to emerge, and the remaining UNLA garrisons had great difficulties in holding out. The insurgents proved better trained and more effective fighters, and often captured supply convoys coming from the south. In addition, the UNLA garrisons suffered from indiscipline and internal rivalries, sometimes clashing in their barracks. Despite these advantages, the West Nile rebellion was crippled by infighting, high corruption, and lack of actual strategy among the rebel leadership. As the UNRF and FUNA fought each other, the former gained the upper hand and mostly evicted FUNA from West Nile by July 1981. FUNA commander Elly Hassan fled to Sudan, where he was eventually arrested by local authorities. Regardless, the inter-rebel struggle only resulted in the overall weakening of the West Nile insurgents. By 1981, four different insurgent factions were active in northwestern Uganda, all of which claimed to have no direct links with Amin. One West Nile rebel group, the so-called "Nile Regiment" (NR) was set up by Felix Onama, a former follower of Obote. As result of its inability to reduce corruption or provide true stability, the UNRF gradually lost the support of local civilians. The Ugandan government exploited the divisions and chaos among the rebels by launching counter-attacks into Western Nile from 1981, where its regular military and "People's Militia" committed numerous atrocities. By December 1981, the UNLA had retaken much of West Nile while encountering little resistance. Thousands of civilians fled to Sudan in response to the UNLA offensive. However, the UNLA failed to permanently dislodge the West Nile insurgents.


title: Ugandan Bush War

text: In contrast, Obote opted for a more conciliatory approach with the Rwenzururu movement. Following negotiations, the Ugandan government signed a peace agreement with the rebel group's leadership in return for payments and other benefits to the latter. Furthermore, Obote granted the Rwenzururu region some limited autonomy.


title: Ugandan Bush War

text: By late 1981, the UNLA was already in a critical situation. Its rapid expansion to over 15,000 troops by December 1981 resulted in a majority of its troops being untrained, badly armed and often unpaid. Corruption became rampant, and great differences emerged between UNLA units. Some, like those that were active in northern Uganda, were given preferential treatment and became relatively reliable. In contrast, the Central Brigade which mostly fought the NRA mostly consisted of barely-trained militiamen. These troops were considered to be "cannon-fodder" by their own commanders. The counter-insurgency operations against the West Nile rebels were thus much more successful than those against the NRA. Overall, the UNLA already showed signs of great strain at this point, and would have probably collapsed without Tanzanian support by the end of 1981. Regardless, UNLA continued to hold the rebels at bay and even scored several major victories. On 23 February 1982, UNLA fended off a large-scale raid by UFM on Kampala, and then managed to inflict high casualties on the routed insurgents. The UFM attempted to reorganize, but retreated into NRA-held areas. It hoped to convince some of Museveni's followers to defect. Instead, a UFM commander defected with a significant stock of weaponry to the NRA, further weakening the UFM.


title: Ugandan Bush War

text: Fighting in the West Nile region occasionally spilled over into Sudan as UNLA troops pursued rebels over the border. This first occurred in April 1982, when UNLA troops crossed the frontier near Nimule and opened fire on a Sudanese Army unit; the Sudanese troops subsequently detained about 20 Ugandan soldiers. In contrast to other belligerents in the area, the Sudanese Army garrisons in south-eastern Sudan were generally well-disciplined and refrained from attacking civilians.


title: Ugandan Bush War

text: By November 1982, the National Resistance Army, Uganda Freedom Movement, Uganda National Rescue Front, and the Nile Regiment had formed an alliance, called the "Uganda Popular Front" (UPF). Exiled politician Godfrey Binaisa was appointed head of the UPF. While being based in London, Binaisa decided to organize an invasion from Zaire to topple Obote. He attempted to enlist the aid of white mercenaries for this plot, but his plans fell through and were revealed when he was unable to pay for the operation. The entire plot discredited Binaisa. In December 1982, John Charles Ogole was appointed new commander of the UNLA's 11th Battalion in Arua. Ogole reorganized his troops, strengthened morale and discipline, and then launched another counter-insurgency campaign against the West Nile rebels. Ogole's tactics proved highly successful, and ousted most insurgents from the West Nile Region within months. Rebel leader Barnabas Kili was also captured. However, the operation also included widespread destruction and massacres at the hands of the UNLA, whereupon 260,000 people fled the area for Zaire and Sudan. This in turn destroyed the "insurgent infrastructure" of UNRF and FUNA, further weakening them. The UNRF was left mostly destroyed following Ogole's offensive, and relocated from West Nile. The group moved its bases in southern Sudan to northern Uganda, where it attempted to rally the Karamojong people to its cause. In the south, the UNLA under chief of staff Oyite-Ojok waged a counter-insurgency campaign against the NRA in the Luwero Triangle which resulted in the "genocidal killings" of thousands of Baganda civilians. Many government troops deployed in the Luwero Triangle belonged to the Acholi people who became widely hated by the southerners.


title: Ugandan Bush War

text: In 1983, the Obote government launched Operation Bonanza, an extensive military expedition of UNLA forces that alone claimed tens of thousands of lives and displaced a significant portion of the population. The blame for the massacres was placed on the people of northern Uganda for supporting the actions of the NRA, which increased the existing regional tensions in the country. The UNLA also defeated the UFM in 1983, destroying its main camps during a coordinated counter-insurgency operation. The UFM suffered another setback when its exiled leadership was scattered during a crackdown in Kenya in July 1983; its leader Kirya was repatriated to Uganda and imprisoned. The group never recovered. Remnants of the UFM, possibly a few hundred militants strong, subsequently formed the Federal Democratic Movement of Uganda (FEDEMU).


title: Ugandan Bush War

text: Up to late 1983, Obote's government had remained relatively stable and in control of most of Uganda thanks to the efforts of chief of staff David Oyite-Ojok. Though it could not defeat the NRA, the military was able to contain it. Despite this, Obote's forces suffered from tribalism, corruption, and internal rivalries. The UNLA and its allied militias had been expanded too quickly in an attempt to defeat the insurgency: By 1984, Obote had 35,000 to 40,000 men under arms, but just 15,000 had received basic training. As a result, the soldiers were undisciplined, unreliable, and prone to harass, steal from, and murder civilians due to a lack of proper pay and supplies. Although the Ugandan government knew that it could not even feed its large army, let alone properly train or arm it, Obote was unwilling to demobilize troops out of fear that the soldiers could behave even worse if they were no longer employed. Despite its massive military and militia support, the government was also unable to fully suppress violence in the northeast, where Karamojong raiders continued to operate. Obote's forces were at least able to contain the Karamojong cattle raiding, keeping the raiders out of other regions.


title: Ugandan Bush War

text: The situation began to change with Oyite-Ojok's death under suspicious circumstances in a plane crash in December 1983. At first, people believed that the chief of staff had been killed by rebels who consequently assumed responsibility. Oyite-Ojok's loyal troops, most importantly the People's Militia and National Youth Army responded by carrying out revenge killings against suspected rebel supporters. After one week, however, rumours spread among the military according to which Obote had arranged the death of his chief of staff due to developing rifts between them. Although Obote's responsibility could not be proven, the rumours damaged Obote's reputation among the military. The CIA determined that Oyite-Ojok had been crucial for keeping the Ugandan government afloat, and had been responsible for "maintaining some semblance of security and order" in the country. With him gone, the UNLA began to unravel. An increasing number of Acholi soldiers believed that Obote was using them as cannon-fodder, while filling the country's leadership with Langi. At the same time, the NRA became more successful at spreading its propaganda, and attracting dissatisfied Acholi army officers to their cause. Foreign support for Obote had also diminished. Besides the still significant North Korean aid, just 50 Tanzanian, 12 British and six U.S. advisors remained in the country.


title: Ugandan Bush War

text: However, the government initially remained relatively successful in battling the rebels. With Oyite-Ojok dead, Obote appointed Ogole new head of the anti-NRA operations in the Luwero Triangle. Ogole improved his troops' training and included other security as well as civil agencies in his anti-rebel strategy. In a series of operations, he managed to mostly oust the NRA from the region, forcing it to retreat west into the Rwenzori Mountains and Zaire. In addition, the Ugandan government decided "to teach the Karamojong a lesson" after raiders had taken advantage of Oyite-Ojok's death to attack his farm and kill over 100 militiamen in the northeast. In cooperation with the Kenyan government, UNLA and allied militias launched a campaign which largely destroyed or seized the Karamojong's food sources in the form of cattle and fields. This left the nomads almost totally dependent on international aid agencies like the World Food Programme and UNICEF for survival.


title: Ugandan Bush War

text: However, the unrest in the UNLA eventually escalated when Acholi troops mutinied in Jinja and other locations in June 1985. Rifts subsequently erupted in the government and some political groups such as the Democratic Party attempted to exploit the chaos by gaining control over the military. The news also reached Gulu, where Lieutenant General Bazilio Olara-Okello, an Acholi, was stationed. Fearing that a new government in Kampala might purge the Acholi, he revolted. Olara-Okello gathered a force dominated by Acholi mutineers, and won the support of ex-Amin loyalists from the West Nile and Sudan. Using these troops, he conquered Lira, and then marched on Kampala. The capital fell after a short battle in July 1985, but Obote had already fled to Tanzania. He later relocated to Kenya and finally Zambia. After the successful coup, General Tito Okello was installed as President; this marked the first time in Uganda's history when Acholi had achieved state power. The coup had catastrophic consequences for the UNLA. The new Acholi leadership promptly began to use their new power to disempower and exploit other ethnic groups including Langi, resulting in the collapse of discipline and order among many military units. From then on, the UNLA gradually devolved into "marauding bands" and declined in numbers to about 15,000 troops by late 1985. Some commanders such as Ogole fled into exile. On 23 August, the 196 North Korean military advisers to the UNLA were flown out of Uganda.


title: Ugandan Bush War

text: Regardless, Okello's government was successful in opening negotiations with several rebel groups, arguing that Obote – their common enemy – had been overthrown. The government reached an agreement with the FUNA, UNRF (I), FEDEMU, and the reactivated UFM. These rebel groups agreed to join a new National Unity government, officially integrated their militias into the government army, and their commanders became part of the ruling military council. Despite this, the ex-insurgents maintained autonomy. Kampala was carved up among the government coalition: UNLA held the center, FEDEMU the south, and FUNA the north. The situation consequently remained volatile, as the government proved fragile, while soldiers and other militants acted with impunity in the capital. Okello also attempted to bolster the UNLA by recruiting large numbers of Karamojong, even though this meant potentially arming cattle raiders. In addition, Okello's government suffered from a lack of respect among the country's elite, as most of its members were uneducated and considered ill-prepared to actually rule Uganda.


title: Ugandan Bush War

text: In contrast to the other insurgents, the NRA refused to compromise with Okello's regime out of ideological reasons. It only agreed to peace talks in Nairobi after international pressure, but never intended to honor any ceasefire or power-sharing deal. Taking advantage of the chaos following Obote's fall, the NRA took full control of the Luwero Triangle as well as much of Uganda's west and south. It also received fresh weapons shipments from Libya and Tanzania. Museveni's group was thus in a position of strength, and used the Nairobi peace talks to stall for time. It even ostensibly agreed to a power-sharing deal. In truth, however, the NRA prepared its force for a decisive offensive.


title: Ugandan Bush War

text: In August 1985, the NRA launched a series of coordinated attacks that resulted in the capture of significant amounts of territory in central and western Uganda. It besieged and captured the crucial garrison towns of Masaka and Mbarara, greatly weakening the UNLA. In course of these operations, the NRA greatly expanded by recruiting new troops in captured territories and absorbing defected government soldiers. In a few months, it enlisted around 9,000 fighters, growing to about 10,000 troops overall by December 1985. That month, a peace agreement was signed by Okello's government and the NRA, but the deal broke down almost immediately as both parties violated the agreed truce. By January 1986, the UNLA was starting to collapse as the rebels gained ground from the south and southwest. Okello's regime effectively ended when Kampala was captured by the NRA on 26 January 1986. Yoweri Museveni was subsequently sworn in as president on 29 January, and the NRA became the new regular army of Uganda. Tito Okello fled to Sudan. Despite this massive defeat, the UNLA attempted to rally once more, and intended to defend its remaining holdings in northern Uganda.


title: Ugandan Bush War

text: These holdouts were led by Bazilio Olara-Okello, who ordered a mass mobilization in Gulu and Kitgum. Everyone who could hold a rifle, including women and girls, was armed and provided with an "ad hoc" training. Meanwhile, the NRA continued its offensive, capturing Jinja by late January, followed by Tororo in early February. At this point, the UNLA attempted one last time to stem the rebel advance. It counter-attacked at Tororo, but was repelled. The NRA proceeded to assault the fortified crossings of the Nile, encountering particularly heavy resistance by the UNLA and allied West Nile militias at Karuma and Kamdini. After bitter fighting, the NRA overcame the UNLA's defenses, inflicting "catastrophic losses" on the Acholi troops. With effective resistance no longer possible, the UNLA disintegrated and its remnants fled into exile, along with many former government officials. The NRA captured Gulu and Kitgum in March 1986, while the defeated Acholi soldiers mostly returned to their villages. The war appeared to be over.


title: Ugandan Bush War

text: It has been estimated that approximately 100,000 to 500,000 people, including combatants and civilians, died across Uganda as a result of the Ugandan Bush War. Overall, Obote's regime proved to be even more brutal and killed more people than Amin's.


title: Ugandan Bush War

text: The NRM's rise to power was initially met by a large portion of Ugandan's population with trepidation and confusion. Most knew little of the NRM, and it was feared that the new government might prove equally incapable and unstable as the previous regimes. After a few months, however, many Ugandans began to view the NRM with approval, as the party actually succeeded in improving stability and restoring order throughout many parts of Uganda. Regardless, Museveni's government quickly faced significant armed opposition. In fact, the NRA had formally won the civil war, but fighting had not stopped in the north. Various anti-NRA rebel groups and remnants of UNLA remained active, with major insurgencies affecting Acholiland and West Nile in particular. The UNLA's Acholi soldiers had never been disarmed, and many had grown accustomed to their lives as soldiers. They were no longer willing to live as peasants, and were dissatisfied with the new government as well as the traditional rule of the tribal elders. Many were extremely poor, and economic and political chaos was widespread in northern Uganda in the Bush War's wake. As time went on, groups of ex-soldiers began to band together as bandits, and violence gradually grew worse in the north. Some NRA garrisons in the region mishandled the crisis by responding with extreme brutality. Though many NRA troops actually behaved well, the army's undisciplined elements tarnished the reputation of Museveni's government. Rumours began to spread that the government was planning to kill all male Acholi. Many Acholi feared that the NRA sought revenge for the mass murders in the Luwero Triangle during the Bush War. In fact, many southerners blamed not just the violence of the Bush War on the Acholi, but even the brutal regime of Idi Amin – even though the Acholi had been marginalized by Amin. The Karamojong ex-UNLA soldiers also took large quantities of weaponry to their home territories after the collapse of Okello's regime, subsequently increasing their raids in scale and numbers. This general unrest contributed to the return of open war to northern Uganda. Over time, the NRM-led government would face more rebellions than both the Amin as well as the Obote regimes, yet survived all of them.


title: Ugandan Bush War

text: Milton Obote never returned to Uganda following his second overthrow and exile, despite repeated rumors he planned to return to Ugandan politics. Obote resigned as leader of the Ugandan Peoples Congress and was succeeded by his wife, Miria Obote, shortly before his death on 10 October 2005 in South Africa. Tito Okello remained in exile in Kenya until 1993, when he was granted an amnesty by Museveni and returned to Uganda, where he died in Kampala in 1996.


title: Ugandan Bush War

text: Despite its support for Obote during the civil war, North Korea quickly developed amicable relations with Museveni's government. Cooperation was restored as soon as 1986, and the new Ugandan military consequently received weaponry as well as training by North Korea. The country's involvement in the Bush War had other long-lasting repercussions, however, as North Koreans became a symbol for mystical military power in northern Uganda. As result, rebel groups such as the Holy Spirit Movement went on to claim that they were aided by North Korean spirits in their war against Museveni's government.


title: Ugandan Bush War

text: The ranks of the UNLA included many ethnic Acholi and Langi, who had themselves been the victims of Idi Amin's genocidal purges in northern Uganda. Despite this, like Amin, the UNLA under Obote targeted and abused civilians. These abuses included the forced removal of 750,000 civilians from the area of the then Luweero District, including present-day Kiboga, Kyankwanzi, Nakaseke, and others. They were moved into refugee camps controlled by the military. Many civilians outside the camps, in what came to be known as the "Luweero triangle", were continuously abused as "guerrilla sympathizers". The International Committee of the Red Cross has estimated that by July 1985, the Obote regime had been responsible for more than 300,000 civilian deaths across Uganda.


title: Ugandan Bush War

text: The NRA also committed atrocities, as land mines were used against civilians, and child soldiers were widespread in the NRA's ranks, and continued to be after the NRA had become the regular Ugandan army. In the early stages of the war the NRA also executed some government-aligned chiefs and directed its civilian supporters to murder UPC activists. UPC members alleged that the NRA massacred civilians, though NRA leaders denied this.


title: Ugandan Bush War

text: The Ugandan Bush War was depicted in the 2018 film "27 Guns". It was written and directed by Natasha Museveni Karugire, Yoweri Museveni's eldest daughter.


title: Linear differential equation

text: In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form


title: Linear differential equation

text: where and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable .


title: Linear differential equation

text: Such an equation is an ordinary differential equation (ODE). A "linear differential equation" may also be a linear partial differential equation (PDE), if the unknown function depends on several variables, and the derivatives that appear in the equation are partial derivatives.


title: Linear differential equation

text: A linear differential equation or a system of linear equations such that the associated homogeneous equations have constant coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is also true for a linear equation of order one, with non-constant coefficients. An equation of order two or higher with non-constant coefficients cannot, in general, be solved by quadrature. For order two, Kovacic's algorithm allows deciding whether there are solutions in terms of integrals, and computing them if any.


title: Linear differential equation

text: The solutions of homogeneous linear differential equations with polynomial coefficients are called holonomic functions. This class of functions is stable under sums, products, differentiation, integration, and contains many usual functions and special functions such as exponential function, logarithm, sine, cosine, inverse trigonometric functions, error function, Bessel functions and hypergeometric functions. Their representation by the defining differential equation and initial conditions allows making algorithmic (on these functions) most operations of calculus, such as computation of antiderivatives, limits, asymptotic expansion, and numerical evaluation to any precision, with a certified error bound.


title: Linear differential equation

text: The highest order of derivation that appears in a (linear) differential equation is the "order" of the equation. The term , which does not depend on the unknown function and its derivatives, is sometimes called the "constant term" of the equation (by analogy with algebraic equations), even when this term is a non-constant function. If the constant term is the zero function, then the differential equation is said to be "homogeneous", as it is a homogeneous polynomial in the unknown function and its derivatives. The equation obtained by replacing, in a linear differential equation, the constant term by the zero function is the "associated homogeneous equation". A differential equation has "constant coefficients" if only constant functions appear as coefficients in the associated homogeneous equation.


title: Linear differential equation

text: The solutions of a homogeneous linear differential equation form a vector space. In the ordinary case, this vector space has a finite dimension, equal to the order of the equation. All solutions of a linear differential equation are found by adding to a particular solution any solution of the associated homogeneous equation.


title: Linear differential equation

text: A "basic differential operator" of order is a mapping that maps any differentiable function to its th derivative, or, in the case of several variables, to one of its partial derivatives of order . It is commonly denoted


title: Linear differential equation

text: in the case of functions of variables. The basic differential operators include the derivative of order 0, which is the identity mapping.


title: Linear differential equation

text: A linear differential operator (abbreviated, in this article, as "linear operator" or, simply, "operator") is a linear combination of basic differential operators, with differentiable functions as coefficients. In the univariate case, a linear operator has thus the form


title: Linear differential equation

text: where are differentiable functions, and the nonnegative integer is the "order" of the operator (if is not the zero function).


title: Linear differential equation

text: Let be a linear differential operator. The application of to a function is usually denoted or , if one needs to specify the variable (this must not be confused with a multiplication). A linear differential operator is a linear operator, since it maps sums to sums and the product by a scalar to the product by the same scalar.


title: Linear differential equation

text: As the sum of two linear operators is a linear operator, as well as the product (on the left) of a linear operator by a differentiable function, the linear differential operators form a vector space over the real numbers or the complex numbers (depending on the nature of the functions that are considered). They form also a free module over the ring of differentiable functions.


title: Linear differential equation

text: There may be several variants to this notation; in particular the variable of differentiation may appear explicitly or not in and the right-hand and of the equation, such as or .


title: Linear differential equation

text: The "kernel" of a linear differential operator is its kernel as a linear mapping, that is the vector space of the solutions of the (homogeneous) differential equation .


title: Linear differential equation

text: In the case of an ordinary differential operator of order , Carathéodory's existence theorem implies that, under very mild conditions, the kernel of is a vector space of dimension , and that the solutions of the equation have the form


title: Linear differential equation

text: where are arbitrary numbers. Typically, the hypotheses of Carathéodory's theorem are satisfied in an interval , if the functions are continuous in , and there is a positive real number such that for every in .


title: Linear differential equation

text: where are (real or complex) numbers. In other words, it has constant coefficients if it is defined by a linear operator with constant coefficients.


title: Linear differential equation

text: The study of these differential equations with constant coefficients dates back to Leonhard Euler, who introduced the exponential function , which is the unique solution of the equation such that . It follows that the th derivative of is , and this allows solving homogeneous linear differential equations rather easily.


title: Linear differential equation

text: be a homogeneous linear differential equation with constant coefficients (that is are real or complex numbers).


title: Linear differential equation

text: Searching solutions of this equation that have the form is equivalent to searching the constants such that


title: Linear differential equation

text: When these roots are all distinct, one has distinct solutions that are not necessarily real, even if the coefficients of the equation are real. These solutions can be shown to be linearly independent, by considering the Vandermonde determinant of the values of these solutions at . Together they form a basis of the vector space of solutions of the differential equation (that is, the kernel of the differential operator).


title: Linear differential equation

text: In the case where the characteristic polynomial has only simple roots, the preceding provides a complete basis of the solutions vector space. In the case of multiple roots, more linearly independent solutions are needed for having a basis. These have the form


title: Linear differential equation

text: where is a nonnegative integer, is a root of the characteristic polynomial of multiplicity , and . For proving that these functions are solutions, one may remark that if is a root of the characteristic polynomial of multiplicity , the characteristic polynomial may be factored as . Thus, applying the differential operator of the equation is equivalent with applying first times the operator and then the operator that has as characteristic polynomial. By the exponential shift theorem,


title: Linear differential equation

text: As, by the fundamental theorem of algebra, the sum of the multiplicities of the roots of a polynomial equals the degree of the polynomial, the number of above solutions equals the order of the differential equation, and these solutions form a base of the vector space of the solutions.


title: Linear differential equation

text: In the common case where the coefficients of the equation are real, it is generally more convenient to have a basis of the solutions consisting of real-valued functions. Such a basis may be obtained from the preceding basis by remarking that, if is a root of the characteristic polynomial, then is also a root, of the same multiplicity. Thus a real basis is obtained by using Euler's formula, and replacing formula_16 and formula_17 by formula_18 and formula_19.


title: Linear differential equation

text: If and are real, there are three cases for the solutions, depending on the discriminant . In all three cases, the general solution depends on two arbitrary constants and .


title: Linear differential equation

text: Finding the solution satisfying and , one equates the values of the above general solution at and its derivative there to and , respectively. This results in a linear system of two linear equations in the two unknowns and . Solving this system gives the solution for a so-called Cauchy problem, in which the values at for the solution of the DEQ and its derivative are specified.


title: Linear differential equation

text: where are real or complex numbers, is a given function of , and is the unknown function (for sake of simplicity, "" will be omitted in the following).


title: Linear differential equation

text: There are several methods for solving such an equation. The best method depends on the nature of the function that makes the equation non-homogeneous. If is a linear combination of exponential and sinusoidal functions, then the exponential response formula may be used. If, more generally, is a linear combination of functions of the form , , and , where is a nonnegative integer, and a constant (which need not be the same in each term), then the method of undetermined coefficients may be used. Still more general, the annihilator method applies when satisfies a homogeneous linear differential equation, typically, a holonomic function.


title: Linear differential equation

text: where is a basis of the vector space of the solutions and are arbitrary constants. The method of variation of constants takes its name from the following idea. Instead of considering as constants, they can be considered as unknown functions that have to be determined for making a solution of the non-homogeneous equation. For this purpose, one adds the constraints


title: Linear differential equation

text: Replacing in the original equation and its derivatives by these expressions, and using the fact that are solutions of the original homogeneous equation, one gets


title: Linear differential equation

text: This equation and the above ones with as left-hand side form a system of linear equations in whose coefficients are known functions (, the , and their derivatives). This system can be solved by any method of linear algebra. The computation of antiderivatives gives , and then .


title: Linear differential equation

text: As antiderivatives are defined up to the addition of a constant, one finds again that the general solution of the non-homogeneous equation is the sum of an arbitrary solution and the general solution of the associated homogeneous equation.


title: Linear differential equation

text: The general form of a linear ordinary differential equation of order 1, after dividing out the coefficient of , is:


title: Linear differential equation

text: where is an arbitrary constant of integration and formula_35 is any antiderivative of . Thus, the general solution of the homogeneous equation is


title: Linear differential equation

text: For the general non-homogeneous equation, one may multiply it by the reciprocal of a solution of the homogeneous equation. This gives


title: Linear differential equation

text: where is a constant of integration, and is any antiderivative of (changing of antiderivative amounts to change the constant of integration).


title: Linear differential equation

text: A system of linear differential equations consists of several linear differential equations that involve several unknown functions. In general one restricts the study to systems such that the number of unknown functions equals the number of equations.


title: Linear differential equation

text: An arbitrary linear ordinary differential equation and a system of such equations can be converted into a first order system of linear differential equations by adding variables for all but the highest order derivatives. That is, if appear in an equation, one may replace them by new unknown functions that must satisfy the equations and for .


title: Linear differential equation

text: A linear system of the first order, which has unknown functions and differential equations may normally be solved for the derivatives of the unknown functions. If it is not the case this is a differential-algebraic system, and this is a different theory. Therefore, the systems that are considered here have the form


title: Linear differential equation

text: The solving method is similar to that of a single first order linear differential equations, but with complications stemming from noncommutativity of matrix multiplication.


title: Linear differential equation

text: Its solutions form a vector space of dimension , and are therefore the columns of a square matrix of functions , whose determinant is not the zero function. If , or is a matrix of constants, or, more generally, if commutes with its antiderivative , then one may choose equal the exponential of . In fact, in these cases, one has


title: Linear differential equation

text: In the general case there is no closed-form solution for the homogeneous equation, and one has to use either a numerical method, or an approximation method such as Magnus expansion.


title: Linear differential equation

text: A linear ordinary equation of order one with variable coefficients may be solved by quadrature, which means that the solutions may be expressed in terms of integrals. This is not the case for order at least two. This is the main result of Picard–Vessiot theory which was initiated by Émile Picard and Ernest Vessiot, and whose recent developments are called differential Galois theory.


title: Linear differential equation

text: The impossibility of solving by quadrature can be compared with the Abel–Ruffini theorem, which states that an algebraic equation of degree at least five cannot, in general, be solved by radicals. This analogy extends to the proof methods and motivates the denomination of differential Galois theory.


title: Linear differential equation

text: Similarly to the algebraic case, the theory allows deciding which equations may be solved by quadrature, and if possible solving them. However, for both theories, the necessary computations are extremely difficult, even with the most powerful computers.


title: Linear differential equation

text: Nevertheless, the case of order two with rational coefficients has been completely solved by Kovacic's algorithm.


title: Linear differential equation

text: Cauchy–Euler equations are examples of equations of any order, with variable coefficients, that can be solved explicitly. These are the equations of the form


title: Linear differential equation

text: A holonomic function, also called a "D-finite function", is a function that is a solution of a homogeneous linear differential equation with polynomial coefficients.


title: Linear differential equation

text: Most functions that are commonly considered in mathematics are holonomic or quotients of holonomic functions. In fact, holonomic functions include polynomials, algebraic functions, logarithm, exponential function, sine, cosine, hyperbolic sine, hyperbolic cosine, inverse trigonometric and inverse hyperbolic functions, and many special functions such as Bessel functions and hypergeometric functions.


title: Linear differential equation

text: Holonomic functions have several closure properties; in particular, sums, products, derivative and integrals of holonomic functions are holonomic. Moreover, these closure properties are effective, in the sense that there are algorithms for computing the differential equation of the result of any of these operations, knowing the differential equations of the input.


title: Linear differential equation

text: A "holonomic sequence" is a sequence of numbers that may be generated by a recurrence relation with polynomial coefficients. The coefficients of the Taylor series at a point of a holonomic function form a holonomic sequence. Conversely, if the sequence of the coefficients of a power series is holonomic, then the series defines a holonomic function (even if the radius of convergence is zero). There are efficient algorithms for both conversions, that is for computing the recurrence relation from the differential equation, and "vice versa".