I paused it while we were chatting. That's cool. Dual degree VA mall program, I was at Tufts and I was out of school about 10 years, originally when I left Tufts and Fletcher, all of my career has been in the international nonprofit sector and I worked largely in monitoring and evaluation, research for common ground, doing evaluation work across the Middle East and North Africa and elsewhere. For the last few years, I've been living in the United States doing various types of nonprofit work. I just moved to the Seattle area and right now I work for the WCMedia Foundation, which is the organization behind WCPDF where I do sort of global community management work to make WCPDF more diverse than it is. That's me. Great. Lizzie. Yeah, so I was, I see a message that I think we are having some issues here in Kelsey. I don't know if that's the same issue for me now or if. No, you're loud. I could hear Kelsey, but maybe there were a couple. Could you guys hear Kelsey? I was also having, she sounded very quiet. I could still hear her though. Maybe. Okay. All right, Lizzie, where did you go? So my name is Lizzie Robinson. I graduated from Tufts in 2015 and did EPIC my junior year, pretty much a few weeks after I graduated from Tufts, I moved abroad and haven't really moved back to the US since. So I'm currently in Jordan. I've been here for three years now, I've previously worked in also Iraq and Afghanistan. I'm currently the Asia regional lead at a company called Magenta that focuses on the social and behavioral change communication, which essentially means we look at how to improve behaviors and social norms in the development sector. So I have not gone to grad school yet. I believe the other three panelists have. So I will be kind of advocating for the why you should not go to grad school now perspective. Elaine. Hi, everybody. Can you hear me okay? Great. So my name is Elaine Stucker. I graduated from Tufts in 2014. I also did EPIC with Lizzie. So that was my senior year among a bunch of other IGL programs. After I graduated, I worked at the Fletcher School for three years. And then I started my master's in international relations at the University of Chicago, which I took an extra year on and finished in 2019. And then I matriculated into my current PhD program at UCLA. I'm pursuing a master's in statistics and a PhD in political science. And before that, I did market research consulting, which I still continue to do. So I've kind of walked two different tracks simultaneously, which I don't know if I would necessarily recommend for all people, but it was really important to me that grads will not close any doors and just open them for me. So this has kind of been the path I've taken. And Lou May. Hi, everyone. Good morning. Good afternoon. Good evening. Hello. Can you hear me now? Yeah. Okay. Thanks for having me. Lou May, Wong Murphy graduated 2011 from Tufts. I did EPIC in 2010 when it was the city's theme, still very salient through the IGL is how I met my husband actually, Pan Murphy. And I'm currently, I'm a corporate strategist at ABN Bev. So we are the largest CPG company by EBITDA. You probably know us by some of our brands like Corona, Stella Artois, Budweiser. Previously, at the very beginning, I started off my career on Capitol Hill. So worked for Senator John Kerry and Congressman Scott Peters on climate change, agriculture, energy policy. I went to grad school. I pursued my MBA from Kellogg at Northwestern, and it was to make a career change. And after that, I was a strategy consultant at Deloitte, and I'm a corporate strategist at ABN Bev. So I'm happy to talk about business school, if it's right for you or not, and when to go. It's very much a personal decision and look forward to speaking with you guys. Great. Great. All right. Why don't we start with Kelsey, do you want to talk about kind of your decision to go to grad school?  And how that's kind of played out for you now? Yeah, can you hear me okay this way? Yeah, yeah. Great. So I actually wouldn't recommend the course that I took to graduate school, going in straight from being an undergraduate. I honestly, I did, why did Epic in 2007, 2008, so a long time ago, but I really wanted to have a career in the type of work that I was studying in Epic. I wanted to do, you know, work in basically global conflict studies. And so I thought that the Fletcher School would be the best route for that. And that's the kind of things that I studied in graduate school. But what I really come to learn over the sort of intervening years is that a lot of the, I think, more thematic and area study courseworks that I did instead of getting like a straight professional degree, like business, law, medicine and so forth, has been a lot harder to leverage career wise, especially if you don't live in DC, or you are committed to having a career abroad for sort of the rest of your life. And so it's been a little bit of an inflexible choice that I've found a lot of ways to sort of like hack and be creative and try different opportunities that I'm happy with where I am right now. But that's not a pathway that I would recommend for someone unless you're really interested in like academia or you have a lot of control of the way you live. What areas did you focus in when you were at Fletcher? Yeah, well, so the best choice that I made at Fletcher was I studied monitoring and evaluation and that was a really hard skill set and I worked in that field for a number of years and I just moved away from it a little bit more recently. So that's a good trap. But other than that, I did like political theory studies, conflict studies, I did commubiation coursework and that's just been a lot harder to make use of. Gotcha. All right, Elaine, you also went to, while you worked at Fletcher and then you went to grad school pretty soon after. So do you want to talk about that? Sure. So like we may mentioned, I went to grad school kind of to pivot into a new field or at least a new position. I saw it as a way for me to get out of what was kind of becoming a dead-end track for me. Well, I loved working at the Fletcher school. I ended up like doing so many other things on the side to try to supplement what I was doing. Like I was an RA for various professors at Fletcher. I also worked on a negotiation project with a professor and a mediation project and then I also, what else did I do? I worked as an RA for a Harvard project. So I felt like I was just kind of scrambling to find meaning in what I was doing. And so then it made sense to me that one, I wanted to like reorient myself and like have kind of like a fresh start after an MA, which is why I originally started pursuing an MA. And I also wanted to learn like some harder skills as Kelsey was saying. So because I was employed at the Fletcher school, they let me take classes. So I learned programming, like computer programming. And then when I started my master's in Chicago, I also like continued to learn programming and started learning math because I really wanted, like Kelsey said, I pretty much am an agreement. I wanted to learn a harder and more concrete set of skills that I could point to and say like not only do I have these analytical skills, but I also can functionally like write code and do regressions and defend this causal interpretation of what I'm doing. And so the biggest reason that I went to grad school when I did was I got a full scholarship and I have really encouraged people not to go to grad school unless it's funded and hopefully fully funded. I think that it would have been unbelievably difficult to not have funding and still be paying off my undergrad loans. And I did work throughout my master's and I'm still working throughout my PhD. But the work I do now is to supplement my stipend instead of to try to make ends meet. And so it's just a whole different ball game in terms of stress when you have funding. So I did get a full ride to Chicago. And I think the little things I was doing along the way like that RA work I mentioned and working on these projects extracurricularly did help me get funding. But I also only applied to programs that supported their students. So I did get in other places and some places were in DC and I did actually get into Fletcher. But like you have to be aware of how much aid they're able to offer you and how much you know you're willing to kind of give up on that funding in order to go to a more prestigious school. But I think University of Chicago is not bad. So that's where I ended up. And then again I'm fully funded here in LA. So a lot of my decisions have been based around funding almost exclusively. I did to know I did three rounds of applications to grad school until I accepted one that met with both like the funding I needed the location I wanted the reputation the school had to hopefully propel me into where I wanted to go. So a lot of my timing was based on money. Great. And Lou May the choice to go to business school I think as most people don't go directly from undergrad mostly because the schools themselves look for practical experience before one applies for an MBA. That means that I think the best advice I received in whether I should pursue a law degree and MPA MPP or business degree came from one of my former bosses. And at that time I was thinking hard about trying to get a master's in public policy. And it was actually a legislative director for John Kerry and he pulled me aside and he said you know it's about opening up your future options. And at that time back in 2014 I think context is everything at that time in 2014 business school was going to open a lot more options for me. So that being said I think for all of you it's also taking me to account your context. You know what is it in a post COVID world. I think about Patton my husband. He went to grad school a year after he graduated in the middle of the last financial recession. And so there are lots of trade-offs one will make in thinking about going to grad school whether it's to accelerate your career make a career change pursue a field of study that you absolutely love and are passionate about or perhaps also to you know kind of ride out certain things happening in the wider world but any case MBA better to better to work a couple years and then go to it. And Lizzie. Yeah so I I worked basically exclusively in international development since I graduated. So I guess my perspective is really focused on that sector. What I've found is that there's a perception that it can be very hard to break in to kind of the international NGO sector which which I agree it's hard. I think it's hard to break into every any sector. But I think particularly with with this field of work people feel the need to get a master's in order to break in which is has really been contrary to my experience and in my experience with applying to jobs and also recruiting for positions within my current organization and other organizations experience matters so so much more than having a degree. And I think a lot of people either again they they kind of don't know how to break in they don't know how to get that first job. They they're you know they're not quite sure how to make that transition from maybe DC to abroad. And they feel like a master's can be a track to that. And I mean it could be I'm not going to say a master's is is never you know it's not going to be ever a good decision. But I think especially in the when you're kind of just starting out an international development development I think there's so much more to learn from jobs and experience and from getting a master's I've worked with people who have gotten master's in you know international development international economic policy whatever it is kind of in their their early and mid 20s and they still they still don't seem to know what they want to focus on. And I think if you're going to dedicate you know one or two years of your life to a grad degree and you know quite a bit of money. Again kind of in the sector of international development I really think it's helpful to have an idea of kind of where you are currently in your career where you want to go and how that degree is specifically going to help you get there. Otherwise I think again experience is much more valuable than a master's. I mean just I think there is a huge gap between kind of what academia is going to teach you and what you're going to learn you know quote in the field. So again I would just I would really encourage people to do try to get experience even if it's frustrating even if it's difficult which it always is. This is never easy but I I don't think getting a master's say in your earlier mid 20s is a good way to transition international development. I'm personally waiting kind of another few years and then we'll probably as some of the other panelists have said we'll probably use my master's to pivot a little bit towards the education sector again within international development. But that's you know after I've had five and probably it'll be you know seven years of figuring these things out and thinking through what I really want to do before kind of taking that step back and getting a master's. No wonder if each of you could Lizzy you brought up the experiential aspect if you guys can talk about kind of the experiential aspect that might have gone along with grad school for Kelsey, Elaine and Lumay. You know like what were you able to do was that helpful piece? And as a second question is how did your majors at Tufts play into your decisions for grad school? I'm happy to to answer that. So Kelsey do you want to start? Oh sorry. That's fine. Can you still hear me alright? Yeah yeah. Great interest the experiential aspect so the mall program at Fletcher is a two-year master's so the experience piece and there's internships you can do of course while you're studying but the main one is December in between and I interned at the Asia Foundation in Sri Lanka. I have found and this is true for me as an undergraduate and a graduate student that international nonprofits give really crappy internships. I mean and people really I mean I work in that sector now right are just generally too busy and scrambling too much within the sector there's not like a lot of because of because a lot of organizations are short on money they don't have as much budget comfort to do advanced planning and that often puts people in the work position of just sort of working in a short-term way and that limits employers abilities to build really meaningful projects. So I usually had internships where someone would come up with a report from you to write so they could ignore me at a desk for the three months that I was there. So I don't think that that was particularly helpful honestly and I agree very much with everything that Lizzie said because she sort of made the choice that I wish that I had done which is like just focus on experience rather than try to get a master's in what I was already doing as an undergraduate with this international relations and being justice to you. Great. Elaine? Sure so like I mentioned or alluded to I think that being really intentional with the kind of coursework you're taking and the kind of skills that you're trying to acquire was important for my experience in grad school. Like I went into grad school knowing that I wanted to come out with like I think three goals and I was like everything I did had to move me towards one of those goals and one of them was learning programming. One of them was like making myself more attractive to consultancy firms because I thought that was something I wanted to do and I did end up doing it. So that worked out and I also really wanted to network in grad school. I felt like I really didn't network well at Tufts. I triple majored. I graduated you know Magna cum laude. All I did was school and then I got out of school and I had nothing really. I turned down the first job. I got offered for extraneous reasons and I realized that I hadn't networked. I didn't really know anyone and I didn't know how to like Lizzie was saying I didn't know how to break into any field in any sector in particular. And so that was like a really harsh wake up call as a young person scrambling to find a job. And so when I went back to school I was really really intentional about networking. I would say most of my best experiences came not from my classes but from the opportunities that being in a school system afforded me. So being linked to organizations. So there is an organization like the IGL at Chicago and I linked up with them and that was really cool. I got to meet Obama. Like that was like one of the best parts of my experience there. And then learning hard skills being able to you know say now that I'm very proficient in R. I know Python. I know theta. I know a bunch of very overlapping skills at this point. But that was a really important part of my experience. And then just being really intentional. Like I think it's so easy to see grad school as like the logical next step especially when as I'm sure everyone here is you're a high achieving student. You're very smart. You're going to have really great opportunities. And just making sure that you're using those opportunities and kind of like milking them for all that they're worth in terms of getting you towards where you want to be going. And I echo everyone else if you're not sure where that is maybe take a couple years see what it is that you're liking and disliking in your career. And then pivot from there. Great. Rume? So I will say I loved my business school experience. And I think this is a common thread that you find across MBAs and for three reasons. The classes, the people, and then the internship experience. So for the first one the classes it's an extremely practical curriculum. And it's all case studies. You're just reading and discussing what other people and organizations have what decisions they made and why. It's really studying decision making. And the professors are fantastic at any top school and all of you if you applied would get into top schools. And it's very dynamic. It's no one spends all their time studying, but the classroom is intense and focused and you get a lot out of it. To the people are extremely smart, engaging, and you just end up forming amazing bonds with people. You end up traveling to all sorts of places. I mean it's just really a fun time and a good time and a lot of those bonds last beyond school. And then thirdly the internship experience. I think this is, you know, if you are to pursue an MBA this is really the reason why is because of the connection with employers. So you're not going to go to and pursue an MBA if you want to work at a particular small startup. That's not really the best use of your time and resources. But if you're seeking to try to make a career change, work at Amazon, Walmart, a big consulting firm, a PVC firm, whatever it is, the internship for the most part extremely structured. The employers come on campus. So you have that access and it's extremely organized. And so for those three reasons, like for me, MBA was right and I loved my experience. And what was your major as an undergrad? Oh, art history and international relations. So yeah, so I mean one of the questions is how much does your undergrad degree kind of lead you to the next step? How do you feel like that happened? Like to where you are now, not necessarily to grad school, but to where you are now. Lizzie, do you want to go there? Yeah, so actually two points. I absolutely agree with Kelsey that internships and international NGOs absolutely suck. This is one of my biggest pet peeves of the industry and something that I just, I really think there should be more emphasis on improving for all of the reasons that she said. I actually also interned with the Asia Foundation in DC, which was like kind of mediocre. I've had worse. In terms of course work, I think it's important not to let your undergrad degree kind of put you in a box in terms of where you want to go with grad school. I've seen so many cases where people do whatever in undergrad somehow end up working in international development and then go on and get a master's in something much more development oriented while their BA had nothing to do with that. I think also it's worth thinking outside the box. I think because we come from the system of US universities and kind of, you know, how things worked at Tufts, there could be an incentive to find it like a similar master's program, maybe. I would encourage everyone to take a look at universities outside the US specifically in the UK. I have done a bit of research on grad programs kind of in the international development sector and have found that in the UK the course offerings are a lot more specific. So for example, they'll have a specific master's degree in gender and education in international development, which is just far more specific than you would find in the US. I think a lot of the programs in the US are kind of your general MPP. So I've just found there's a lot more kind of options for specialization from universities outside the US. Plus, no GRE, they're cheaper and they're one year programs. So you're, you know, a lower opportunity costs in terms of not working for a year. Great, great. Kelsey, do you want to talk? Sorry, what was the original question this one? Well, I guess part of it's going into how your undergrad degree do both your choices, but what you're doing now. Like, you know, so you kind of understand like the shift that may happen after. Sure. So because I rolled straight from my undergraduate degree into my graduate degree and they were basically all the same type of studies and coursework, like there wasn't too much pivoting going on and it has had me working in international nonprofits, which I've done and I've worked for some big ones and some great ones, but it's sort of been more of a process of like dealing on the horse and then off the horse. Like when I'm able to live in a location and effectively leverage my network and my skillset to get an awesome international development job than I have one. And then when life makes me move somewhere else, for example, I just moved to Seattle because my husband is a medical resident at the University of Washington. Like every time I move or you have a life change you can't control. I've been unemployed for like a period of a year or so before I find a new a new role. So I like what other panelists have said about finding a grad program that just opens a lot of options because it has been a little bit harder to pivot. I do think like I'm finally old enough and long enough out of school that I have enough experience to leverage moving forward and I don't really need to talk about my graduate school experience because I've done enough. But yeah, it was all pretty linear and I wouldn't recommend doing it that way. Elaine? Sure, I think that. Sorry. What were your three majors? I are Arabic and Middle Eastern studies. I would say that more than the degrees I've talked about like the experiences that I've had. So kind of lumping this back into the experiential question. Like people were a lot more interested in like, oh, you like lived in Jordan or like, oh, you traveled to Iraq or oh, you lived in Nicaragua. Like what's up with that? And like, yeah, I had an Arabic major but like it was far more useful to be able to say like, oh, I know like these people at the University of Jordan or I've done this and that. And I think that's carried through. Like people don't ever ask me about my undergrad beyond like, oh, you did these interesting things. Like, can you tell me about that experience? So it was a lot less about like, the things on my degree and more about like what I did along the way as I was obtaining them. And I think that's definitely held true. Like so many of my opportunities came from doing RA work. Like so I would do my coursework and then I would meet with professors and then they bring me into the projects that they were doing. And I think like that's really the only way to get like meaningful work experience in terms of like understanding if you want to go into academia especially how to create projects and like source funding and get grants and how to like frame things and present them. And you know, just carrying those kind of projects from beginning to end if you're there long enough has been like really far more important than anything. Like I had specializations of my master's. I don't even remember what they were. Like really like my like I said, mining what you're doing for those experiences that are going to make you both a better employee or a better student or a better teacher. And then also like giving you that network like far it doesn't matter what you're studying really if you're able to do those things. Great, you may. I don't think I've talked about my toughest major to an employer for several years now. But I think what has been helpful is that in the process of applying for grad school if you choose to do so. I think you're forced to do a lot of self reflection. And I had to really ask myself like what do I want? In the next three to five years. And in thinking back about my tough experience there are two things that I really took away. One was that being an art history and I are major and especially going through the IGL and Epic. You all are fantastic writers. Fantastic writers way above the skills of many, many, many, many, many people. So take that away. And then two was that I loved I actually really loved studying art history and international relations. And so when it came time to thinking about what type of work I want to do I'm very happy at a global organization. Every day I get to talk about what's happening in Colombia or Mexico or UK or China. And so those I think reflecting back and thinking oh I pursued it because I was passionate about it is helpful. Great. And Lizzie do you want to talk about how you said it was hard breaking in? So what did you have to do to kind of get the internships or get the experience or end up in the position that you're in now? So I was fortunate enough to be able to take an unpaid internship for six months right basically after I graduated which I 100% understand not everyone can do. That's how I did it. I think though what I often tell people is that it just comes down to luck. You and there's so much competition for some of these jobs you know you really just have to be kind of the right person in the right place at the right time and maybe an organization needs to be desperate enough to hire you which is fine sometimes that's you know how it works. So I think Elaine's point about networking is super super important. I'm sure this is true for all sectors especially in international development you need to have someone flag your CV or else there's absolutely no way you were getting an interview. I think also when I was- How did you build your network? So I saw something on Twitter the other day actually that said networking is the worst way to build a network. I really hate like professional networking I think it feels really inauthentic. I mean you have to do it but I think there's a way to go about it being authentic and being humble about kind of where you're coming from being honest about where you want to go also just being straightforward with people like can you introduce me to your colleague who works in the livelihood sector rather than like dancing around with this really polite language of you know I would be so appreciative. I mean I prefer when people get straight to the point I'm happy to help but you know give me something concrete that I can help you with. I and I think it is really about it is about putting yourself out there reaching out to people asking to be introduced having kind of informal chats with people so that when a job does come up you already have those relationships and can follow up and I think generally people are willing to help everyone is busy you know everyone gets multiple sorts of these requests on a regular basis but I think yeah it does really help to kind of find those people who are more entry points into into the sector. I mean through the alumni network for sure you know ask friends of friends go stalk people on LinkedIn see who they know I mean yeah I'm also happy to speak with anyone on this call if you're interested in international development and it's not easy it takes years but and I mean even like five years out of undergrad I still feel like I don't know anyone which is objectively not true but just there's there's kind of you always feel that way I think it's a slow process it does require being proactive it will always feel like you are not getting anywhere but slowly you will. Great one of the questions I had is about the development of both soft and hard skills analytical skills and hard skills and what you think is important you know as people are thinking about heading out in the next like say you know two to four years into the job market like what are you seeing as important skills to have and if people are double majoring or not double majoring you know is it what would your recommendations be Kelsey do you want to start? Sure before answering that question though I wanted to go back to a point on networking that I actually think is really important to me by far the most successful way to network is once you have your foot in the door at any kind of new place as an employee as an intern or currently while you're at cups is to meet as many people in the context of figuring out what they need and how you can help them and so that's I think by far in a way the vast approach and something that's made me successful in a lot of different roles and makes for an authentic way to connect with and meet people and especially when you're new you can introduce yourself to folks at new teams and departments under the guise of learning but once you learn about them and like what they're looking for and the problems that they can't solve that's a place to make networking really stick and then you can back that up with some LinkedIn stockings if you have that connection once you've left the organization Heather can you repeat your question one more time? Sure I think it's looking at kind of developing like soft skills hard skills what are the things that you find most useful as you kind of start to go out in the job market or looking for internships if you're still in grad school you know and what would be good to focus on you know because sometimes like there's advice so you can major in IR and economics but is it better to do like IR and computer science or yes what's the value bill coming out? Yeah I would definitely recommend pairing IR with anything that's hard skill based I think anything in computer science or like coding and related to sort of technology software development stuff is a great bucket another really good one that's very strongly sort of international development is quant skills quantitative analysis quantitative data collection becoming a state of wizard that sort of thing goes really well another bucket that I see that's really useful is a communication skill set skill set and this is more than writing this is sort of being able to do anything related to marketing press releases like building simple video animations those kind of things that help people communicate what they're doing a lot better I think can go really really far great budgeting also great um Elaine sure um so I agree with Kelsey that I definitely wish that I had diversified some of those majors I wish I had developed some really tangible skills I think that if that's not the track that you want to go down as a student then you should work on a project or experience that you can point to like for example in my master's I worked on a project that had me trying to size how many people like estimate how many people lived under criminal governance in various countries and so I was like using you know like there's no easy way of doing that and so I was using like population statistics and then like how many people like live without toilets and that would make them vulnerable to criminal groups and where these criminal groups operating and stuff like that so then when I was interviewing for the market research consultancy job that I ended up getting they were like well how would you like tell me how many pencils there are in Mexico and I was like oh I know how to do this you look at and I like could basically link what I had done to like another context and so I think that either having a having a skill isn't enough and not to put more pressure on you but like be able to say like this is what I did with that skill and this is how I can apply that like from like this very specific context of like gangs to like this other thing where you know this company L'Oreal wants to know how to sell shampoo and like who are their market and so just being able to translate your skills and kind of reorient them I think is a lot easier if you work on a project from start to finish and you say like this is my deliverable these are this is what I learned from doing this and it's like approaching it that way in terms of like experience and project gaining instead of just like gaining a skill for the sake of having skills if that makes sense yeah absolutely absolutely Lou May I think in the in the beginning of your career having being able to say what those hard skills are is important and of course it's more than just that one line on your CV or it's like I know Microsoft Office right please don't put that on your CV but in throughout the other lines of your CV and I think this goes to Elaine's point and Lizzie's point conveying the specificity of what you've done and the impact of it is far more helpful to me as an employer than writing out the very specific skills that you have so saying you know I've seen on so many CVs people say supported XYZ I don't really know what supported means did you just go to things on the copy paper on the copy machine or are you there actually doing analysis brainstorming whatever it might be so I think being really specific about what your contribution is helps show me I'm looking for people who are resourceful who are creative in their problem solving like what Elaine said right that you'll find other ways of getting to the problem because in real life there's not a prescribed way of getting to the answer and that you are a lifelong learner as well I think there's a great book I recently read is range by David Epstein highly recommend it so for all those generalists out there's for everyone who studied IR or economics it's a real booster in helping you come up with language and convey that how you think is far more impactful than having like an extremely extremely you know specialized specialized skill set and Lizzie so I think in the development sector what I found when I am hiring for you know kind of interns who are 22-23 just coming out of an undergrad program I honestly don't expect them to have the skills that they like the hard skills that they will need for this job I don't expect them to have technical knowledge about behavior change I don't expect them to have contextual understanding of Afghanistan so for me it's really more about the attitude and the mindset and some of those kind of bigger picture parts of who you are so we've all mentioned a few of these things quant skills are super important but for this I would say the bar is like pretty low I don't think you necessarily for like an entry-level job in international development you don't necessarily need to know state up but you need to be able to receive and excel data set and make some sense out of those numbers you know just being comfortable with with numbers and calculations and analysis that's super important writing skills which which the other panelists have have also mentioned and again like I am continuously flabbergasted at the low quality of writing and then in the international development sector so again like the bar is like pretty low your writing does not have to be perfect but like if you've come out of Tufts your writing is probably you know far better than than most people in in the sector some other kind of qualities that I often look for humility recognizing that you as a 22 year old do not know everything and that you are eager to learn and want to learn and recognize that the people around you have things to teach you I think that's really important also I like people who read a lot and read widely I think just come to jobs with a better a better big picture understanding of how the world works I think that's really important and also just kind of an analytical and structured approach to thinking so that when you get a when a problem comes up on a project and you will you have to figure out what to do you can break that problem down into different different components and figure out a structured way to address it in all of our interviews for my company we actually include a logic question and for me personally this question is like if I if I only had to ask one question it would be this question because how whether people get it right and how people approach it and what they do when they get it wrong I think is so much more informative about a candidate than anything on their CV something else that people often ask me about is languages I having studied Arabic for I don't know seven years at this point including at tops and with U.S. government scholarships abroad you know et cetera I I don't think unless you are fluent in a language I don't think it is actually that beneficial to keep studying it what I found is that as a foreigner you are never going to be best positioned to be using that language your local counterpoint parts are always going to be better positioned to be engaging in that language with you know participants and projects et cetera also knowing a language doesn't necessarily mean you you have cultural fluency or that the people in the country see you as they're equal you are always going to be perceived as as a foreigner and that that creates barriers you know kind of separately from from language barriers great um another question we have is about funding and thinking about how how to know which graduate schools pay how to know how to think through funding for grad school especially if you know you have loans coming out of undergrad and you know like Lizzy was that a factor in you not going is it a factor in what you decide next like how do you how do you make those decisions so that you know Elaine had talked about you know finding all the all the funding how do you do that Elaine do you want to start with how you pursued that sure yeah so I don't have like a easy answer for you there's no like database that I know of that tells you I think definitely like talking to people who go to the school like I was in a really privileged position in the sense that I worked at Fletcher so one I knew what kind of money they had and didn't have and two I could talk to students that had gotten into a myriad of schools that were of interest to me about the funding packages that they were offered but like I said like I did multiple rounds of applications and I ended up getting like I got two or three I think offers that were like pretty significant funding but at schools that weren't like of the caliber I was looking for and it was hard you know you have to kind of adjudicate between the time cost of like oh like I'm going to do this again to try to find a little bit better funding at a little bit better school and like I said like I was employed I was really fortunate that I had the opportunity to kind of take this time and really think about it but yeah definitely talking to people who have either gone through the application process or are currently enrolled at those schools going to places like so I mentioned that there's many places like the IGL none exactly like the IGL don't get me wrong but going going to places that have like little buckets of money wherever you end up going so I was able to get summer funding through these like various organizations that were at the school but not under the umbrella of the school funding so that's really important as well kind of like put your eggs in some other baskets besides the normal channels that funding comes through also I didn't do this but you can have employers that will fund your masters and maybe Lu May could talk to this because I know it's more common in the business school sector I think but look for opportunities where your employer and Lizzie was also saying it's like your employer is going to want to develop your skills if you don't have them and so maybe try to find someone that is willing to invest in you in that way and they will cover part for all of your program that's another way of finding funding but I think the most valuable thing is to make yourself valuable and that's why I was doing you know all of those research assistant projects and like networking in that way and like really putting my name out there and getting a stamp on projects and getting in those like authorship lines and those like credentials and people thanked me for my contributions and I was able to point to that and say like I am about like I'm a value enhancing student and when I got funding packages I was able to negotiate them because I was able to say like since I submitted my application I've done this, this and this and I improved this, this and this and you know I've published something here and I wrote an op-ed there and like being able to really point to specifics about like what you're going to bring to the table and this is really coming like I'm in an academia focused program like if you're going to be a knowledge producer and you're going to be a professor you need to be able to point to these things and say I'm able to produce knowledge like I'm able to mentor I'm able to teach and that's really what's going to secure you money is saying like if you pay for me to be in your program I'm going to pay back your individence later because I'm going to be an asset to you Does anyone else want to speak to that to the funding aspect? I mean I think Elaine hid it on the nail on the head I think when you oh there you go oh um I think Elaine hid it on the nail on the head right is framing this as an investment in you and basically there's three ways to invest to have you or someone else invest in you you invest in yourself you take out loans or you pay upfront you get your employer to do so I can talk about that or three you get someone else to do it like the school to invest in you right and maybe eventually you pay a little less on your tuition now having an employer invest in you is it happens and it's not just for MBA programs there are consulting firms and and banks out there and all sorts of industries that are willing to invest in their people what comes with that is that afterwards there's usually an expectation that you will provide the expertise and the experience you have gained in graduate school back to to the organization and to the employer and so a lot of times that might be you know you work back there for two years but again I think Elaine has has articulated it very well which is if you're able to say why investing is worth it in you and what then the value add you can bring that helps a lot because for a lot of these employers like Deloitte to get the funding to go to a business school or get an MPH or a grad school degree you have to go through a rigorous process of basically pitching yourself and saying like why you and why this degree and how much more value it'll bring to the organization afterwards does anyone else want to say anything about about that I'll jump in just to add that I wish it was more common in international development for employers to invest in staff specifically kind of funding for grad school I unless you're like quite senior with a very reputable NGO the NGO probably will not do this just unfortunately kind of how NGOs work I actually work for a for-profit company and my employer has kind of like mentioned this option so it's not it's not impossible within international development but probably more common with for-profit entities rather than non-profit great and I guess going to that in terms of developing the skill of kind of pitching yourself or taking that initiative or kind of approaching schools as you know as you have something to add to what they're doing versus you being grateful that they're admitting you you know how do you how do you kind of balance all of that whoever wants to start Elaine yep I think that this is kind of orthogonal to the question but I'll get around to it something that was important for me and I think Lizzie really alluded to this in the sense that you should be humble is that I was really frustrated coming out of my undergrad program like I think I mentioned that I didn't have more opportunities and at first I was frustrated with like everyone else but then in turn like I became frustrated with myself and I think doing that self work the work on yourself rather to like see where you're at and where your gaps are and where you want to be going is so so important to being able to pitch yourself and like having a clear idea of yourself and where you stand and like dealing with whatever baggage or like the things that are preventing you from like going out there and really being able to sell yourself not to get like too like self-help and self-love is really important though like you're not going to be able to network if you like have no sense of your own worth and your own value and like what you can bring to the table and like that people should listen to you and like want to network with you and so you know like go to therapy do whatever you need to do to be able to say like I'm confident in myself and my abilities and like I'm also aware of my deficiencies or the things that I can't do yet and like going off with that mindset I think is so so important to being able to pitch yourself to being able to talk to people like if you can't talk to people like you got that's like step one learn how to talk to people and to like advertise yourself in a way that makes people want to talk to you Yeah I think I want to come to developing the pitch for yourself Can you guys hear me? Yeah There might be a lag I think it's it's always a more efficient use of your time to develop your strengths and lead with that then say oh I've got all these you know you know trying to close certain weaknesses right so as each one of you have varied experience varied backgrounds and don't at all discount what type of you know growing up experiences you had growing up but I think Elaine is right like coming up with like what is your differentiator your competitive advantage is super helpful for me and also it's it also is somewhat defined by who you're talking to and who you think you might be evaluated against so for me going into MBA there are fewer people who have quote unquote untraditional backgrounds and so me talking about you know coming from the hill that was a huge differentiator even though every day I'm surrounded by people on the hill and I didn't feel that special but being able to talk about it and it was very different from other people's day-to-day experience was kind of kind of my value at right I could talk about how policy decisions were made whereas others were not exposed to that so I think you know always thinking about what are your strengths and then matching that with the context in which you're pitching yourself will help you find in that particular moment how to pitch yourself your pitch is going to vary all the time depending on who you're talking to or what you're trying to what you're trying to get and so as long as you have a general story of yourself and you can flex up or down certain experiences that will that helps a lot yeah and building on what Lou may say because I think that's a really good point and feel free to also change how you tell your story and your narrative and how you frame your experience depending on the context or the job or the person and if you don't feel like you have an overarching narrative to your life because most of us don't feel free to make it up I think the so the example I often give is from my own experience so in like the year and a half after I graduated I was moving I lived in three countries each for six months have three different jobs with three different employers one way to look at it at that which I did at the time was that I had no idea what I was doing I was bouncing around randomly I could not hold out a job or I could frame that as I was trying to diversify my contextual understanding of the broader Middle East and Asia region and experimenting with different types of employers to figure out what environment I worked best in so I think it and like I mean there's you know this old old phrase fake it till you make it I mean again don't be arrogant but like you know do what you need to do to sell yourself and then so going back to Elaine's point also about being interesting I think the flip side of that is also be interested in the people that you're talking to people love to talk about themselves this includes your future potential bosses you know hiring managers whoever you're networking with so yes be an interesting person that other people want to talk to but also be interested in other people's experience Kelsey yeah I can add just one quick piece from the international non-profit perspective I think one thing that can be really successful in a pitch that I've used quite a number of times is when you're interviewing if you show that you truly get it and you are a true believer in the mission and focus of that organization so for example when I was interviewing to work at search for common ground which is the world one of the world's largest global peace building organization I went into that interview knowing very very well exactly what peace building is what it tries to change why it can be important in people's lives in a really dynamic way and that can stand out when you're in an interview where they're just talking with tons and tons of people who have IR degrees and are just sending in their resumes any kind of nonprofit assuming that they're all the same or similarly now that I work at Wikipedia I came into my interviews here and I even do this now that I'm an employee where I can really talk about what the free knowledge movement is why Wikipedia is the center of that and why unleashing free knowledge in every language around the world is key to transforming people's sense of empowerment about the world around them and that can be an important differentiator Heather can I add one thing about pitches so whether you're pitching yourself your startup idea whatever it might be an idea I have sat through quite a couple interviews and one mistake that people make no matter if they're just starting out or they're 10 years of experience under their belt is sometimes people's pitches can be very long so time yourself and make sure it is concise brevity is the goal of wit great all right and I have a last question and this is specifically for Elaine and Lou May but Kelsey this may have applied to you when you were thinking about things and you know Lizzie it may apply to you as you're going forward did you consider dual degrees that explore both policy and business such as MBA, MPA or MBA, MPP like did those crop up dual like dual programs you know it could be law and something or you know did you think about that whoever wants to go I'll just say something really quick on that it didn't apply to me directly but I've been very involved in my husband's graduate school decisions and he just finished an MD MPH and Master of Public Health and a medical degree and the way that I would encourage you to look at it is not I want to do a dual degree because I'm interested in two things and like here's the perfect sandwich but instead what is the main degree that you want and is there a specific professional spin on that degree that the extra that the second degree would help you with for example my husband got an MD and an MPP not because he both cares about medicine and public health but he wants to be a physician who takes on public health leadership positions after working as a doctor for a few years and so I think that perspective applies to other dual degree programs in a useful way yeah Kelsey is 100% right so there are quite a couple of combined degrees whether you want at least on the MBA side you can do an MBA, MPH, MBA, MD, MBA, JD, JD, MBA those are quite popular and I think it goes back to at the end of it how are you going to leverage both degrees because it is definitely more work more money and so for example if you pursue an MBA, MD a lot of those folks want to become hospital administrators for example or they want to become senior leaders for scientific research company if you're getting a JD, MBA a lot of those may want to maybe practice certain types of law and so I think it's being very pursuing a dual degree has to be extremely intentional and as long as you're clear on what your intention is then you can pursue it with all the fun rigor that you want Elaine? Yeah, I definitely agree with what's been said as you could probably tell I was like a big fan of just picking up degrees left and right but you really have to have a reason for what you're doing and I didn't really look at dual degrees because I decided that the skill set that I really wanted was like the harder science like computer computer science type things and there really aren't a lot of IR computer science dual degree programs so really what sealed the deal for me with UCLA was the fact that they like about a fourth I think of their cohort right now is able to pursue a master's in statistics through the statistics department here and because I already had a adjacent MA degree that was really of import to me because I didn't want to waste like this opportunity to get a master's in something that I already had a master's in my master's is in IR and I would have been getting a master's in political science as a political science PhD so the fact that this program was able to say like you can actually go outside of our department and pursue this and they also had really close languages with the public policy school like that's kind of how I decided on my PhD program was the other opportunities that afforded me while still being in this like very specific context of what I wanted to be doing in terms of my main study and the other concern I had was that it seemed a lot harder to me and this was a very surface level investigation but I wasn't sure if I could get funded for a dual degree program especially because I only had expertise in one of the fields that I was interested in and so my letters of recommendation were only going to be able to speak to me in that context of international relations or political science and so I worried that someone coming in with a stronger background in law or in business or whatever else I would look at the dual degree program for would have a better narrative and a better set of recommendations to secure any funding that existed around that aspect of the degree program and I didn't want to compete with those people so for all those reasons I didn't look at a dual degree program but I think that if you have a clear and cogent story or reason or rationale behind why you really need both degrees you know like go for it put yourself out there and you can always like as far as I know a lot of those programs they may admit you to just one or the other so you don't really lose anything by applying to both but don't collect skills just for the sake of collecting skills don't collect degrees for the sake of collecting degrees because I think we may said you know it's really time-intensive to do multiple things like that and so if you're not using that time effectively that's taking away from time that you could have been networking or meeting other people in your specific field of interest or getting an internship that's going to land you a job so just be really intentional with your time Great, great Lizzie is there anything you wanted to add to that? Just to say that I guess dual degrees aren't really a thing in international development I mean I've considered doing two back-to-back one-year masters but we'll probably not end up doing that for all of the very valid reasons my fellow panelists have just talked about Great, great All right, well I want to thank you guys for taking the time to talk to us today we really appreciate it and for everyone who's been listening thank you so thank you guys Thank you Thanks Heather Thanks We'll talk to you soon Thanks, take care Bye


Hi everyone, Joe for jazby's casebreaks.com coming at you with 2122 upper deck SP authentic hockey four box random team break number six one spot gets you two teams and There was a free spot giveaway here as well. We're giving away four spots. We only sold 12 spots outright I'm gonna randomize your names top four will get extra spots little buy one get one action Top four after four four the hard way two in a two. Good luck one two three and Four after four John Allen David and Craig Top four after four some extra spots going your way now one spot gets you two. So let's double you up and Let's do the randomizer so you can see all your names right here And oh, yes, perfect. Thank you You too, let's roll it and randomize it Let's roll and randomize it Names and teams five and a six eleven times one two three four five six seven eight nine ten and 11th and final time Craig down at David Craig David Remember him five and a six eleven times for the teams one two three four five six seven eight nine Ten and eleven the final time. You got the Oilers down to the Panthers. There we go Oliver. That's right Seven days. I was a classic filming in Things a little more popular in the UK than than he was here, but All right, so I got Oilers down to Panthers after 11 All right, Craig with the Oilers and the Blues David with the Craig David David with the blue jackets John with the Senators and my Kings David has the Knights Alan with the Sharks and Red Wings Craig with the Maple Leafs David with the Stars Steven with the Ducks and Caps David with the Devils Alan with the Hurricanes John with the Coyotes Victor with the Rangers David with the Flames Craig with the Wild David with the Predators Alan with the Flyers Aaron with the Islanders John with the Kraken Victor with the Canadians Aaron with the Jets David with the Penguins Kevin with the Canucks David with the Abbs and Bruins Kevin with the Sabres Alan with the Blackhawks and Tampa Bay Lightning And David with the Florida Panthers Let's alphabetize by Team and we're in a pause video just for a little bit when we come back We're gonna see if there's any trades and then we'll have the break stick around BRB All right, welcome back everybody. No deals were done So the list remains the same here on a Thursday The next four box break in the store right now ladies and gentlemen check it out. Jasper's case breaks calm And we got Florida and Carolina playing right now Just the end of the first period. I think Carolina is up one nothing Eastern Conference final this is game one, right? Yeah, this is game one Tradiac. What's going on? How are you? Yeah, he is a popular dude.  I guess if that means That means show hey Otani doing good Tradiac Filling some breaks breaking some breaks Watching watching a little playoff hockey, although it's an intermission so I could change back to the Lakers game Watching a little watching my Lakers This is my Friday There's a lot happening here. There's a spectrum FX future watch Tier one that'll be for the Golden Knights. That's gonna go to David. There is a William William Eklund 18 out of one that's kind of hard to read that font to 150 That will be for San Jose. That's gonna go to Craig got awesome Matthews pageantry Conor McDavid Matthew Kachuk die cut and our autograph is a Benoit Olivier Drew for the Ducks 838 out of 999 that's for Stephen K and Anaheim And then we've got a future watch relic and autograph Some cards underneath there for the Rangers Niles Lundquist It's a pretty sharp-looking card 86 out of 99 Who's got the Rangers that's gonna be for Victor? Jack Hughes die cut another box Championship soccer sets next month Nice Yeah, next year Otani Bueller may stone Miller Hopefully Dustin Mays injuries not that bad Maybe one more year Kershaw doesn't Kershaw wants to play with Otani, right? I could see some trades happening in the deadline Oliver people people talking I kind of like the idea of Let's see. Yeah, the White Sox are 16 and 29. They are not playing well. I Would have mind a couple guys on the White Sox. I Would I don't know I think the Brewers are having a better season They might not be willing to part with the Demis and Burns, but what about? Yeah, what about a Tim Anderson Geolito? I'll take a Tim Anderson Geolito. I mean not all those prospects are gonna work out So I mean I'm sure the Dodgers organize a front office figure out who they think they could let go There's another spectrum FX future wash and we've got a Ross Colton for the lightning to 900 That's gonna go to Alan and Tampa Bay The future watch is Yegor Shinakov Tier one for the blue jacket that's gonna go to David B Is Tim Anderson not happy I Imagine he thought he's gonna be on a more competitive team got a Quinn Hughes die cut For Vancouver and it'll be for Kevin and our autograph is Poad Coulson for the Canucks Vancouver Kevin and we've got a sign of the Times autograph Jack Roselovich for the blue jackets That's for David B in Columbus Really? I did not see that video Oliver Good. Yeah, it's not bad, right? I turn I put the game on on my phone. It was ten four Yeah, but then they went up a little yeah Nuggets went up a little bit. I think it's I mean this is a lot better start than last game They're still pushing it on transition against us.  Yeah, finally Finally paying off Oh, they're still down not as not as down as before Yeah, I mean it was a dominant six seven years but His like knees or his back like betrayed him Yeah, I mean that's those lower body injuries are a thing But then you just have genetic freaks like Shaq who's like that big and almost never he played like almost 20 years Right that's kind of weird right that's a bad possession Dennis There's a good foul. Oh, yeah as a as a flagrant tee up that ref right now No Got a limited Leon drives is all 43 out of 99. That's for Edmonton. That'll be for Craig Spectrum FX another tier one Alex Barber lay Future watch for the Lightning Tampa Bay. That'll be for Allen Connor Hell book for Winnipeg That'll be for Aaron die cut and we got a Adam Beckman for the wild 94 out of 9 a 99 Craig with Minnesota That's a flagrant one and we got a Quinten by feel nice future watch autograph for the Kings 763 out of 999 who's got my Kings John does John Watson. There you go. And then we got a Panoran die cut blue For the Rangers it's for Victor All right final box coming up fourth and final we got another four boxes in the store Get after it Jasper's case breaks calm.  They're gonna do that and we got a sign of the times autographed Samuel Gerard For Colorado that's gonna go to David B and the abs All right, I think that's it for the autos, but let's see what else we got here All-card ship and we've got a die cut in the back right there and that's gonna be That's left ski for the lion lions light lightning That's gonna go to Alan and Tampa Bay and there you go ladies and gentlemen pick a random team six in the books I'm Joe for Jasper's case breaks calm and I'll see you next time for the next hockey break. Bye. Bye


Welcome back. In this lecture, we will discuss about the solution of variable coefficient differential equations. We have seen that if the differential equation is constant coefficient whether it is of first order, second order, third order, then the method of solution we have already seen in the previous lectures that you find the characteristic equation solved. After solving or finding the roots, we formulate the solution and we get the general solution and particular solution. And in case if the differential equation is having variable coefficient, then that method fails. For example, if I consider a general differential equation, say a second order. So, consider a second order variable coefficient differential equation, say this is a 0 x y double prime plus a 1 x y prime plus a 2 x y is equal to 0. So, call this equation as 1. Then how to solve it? By the method of characteristic equations and finding the roots will not work. But there are certain situations where special forms of this equation which can be solved just as how we did in a constant coefficient differential equation. So, for example, if we can reduce this variable coefficient differential equation into a constant coefficient differential equation by giving some suitable transformation. For example, if we consider, so consider the Euler Cauchy equation. So, Euler Cauchy equation, so given by x square y double prime plus a x y prime plus b y is equal to 0. So, where a and b are constants. So, this equation though it is a variable coefficient, it is in a nice form. This can be reduced into a constant coefficient differential equation just by applying some transformation. So, consider the transformation, say z is equal to natural logarithm of x or x is equal to e to the power z. So, if you use this transformation, then we are changing the independent variable from x to z. So, if you do this transformation, then what is d y by d x? d y by d x is given by the chain rule, d y by d z into d z by d x, which is equal to by using the transformation 1 by x d z by d x is, z is ln x. So, d z by d x is 1 by x 1 by x d y by d z. So, therefore, differentiating a function with respect to x is equivalent to multiplying 1 by x and taking the derivative of the function with respect to z. That becomes the rule. So, if for this case, then if you find the second derivative, so d square y by d x square is nothing but d by d x of d y by d x. So, we apply the rule stated just above that is differentiating a function, differentiating a function with respect to x is equivalent to 1 by x into differentiating the function with respect to z. So, therefore, we get the second derivative d square y by d x square is equal to d y d x of d y by d x, which is equal to 1 by x times d y d z of d y by d x. So, which is again by definition, 1 by x times d y d z of, what is d y by d x? d y by d x is 1 by x again, 1 by x d y by d z, so which is equal to 1 by x times d y d z of 1 by x is by transformation e to the power minus z, so d y by d z. Now, we can differentiate with respect to z. So, this gives me 1 by x into minus e to the power minus z, so applying the product rule d y by d z plus e to the power minus z times d square y by d z square. So, this is equal to and this e to the power minus z, e to the power minus z is 1 by x and this is also 1 by x, can take 1 by x outside, so this gives me 1 by x square into minus d y by d z plus d square y by d z square. So, therefore, the second derivative d square y by d, d square y by d x square is 1 by x square into minus d y by d z plus d square y by d z square. So, this implies that x square into d square y by d x square is equal to minus d y d z plus d square y by d z square.  And therefore, if we simplifying this and taking d square y by d z square first, so this is d square y by d z square plus a minus 1 d y by d z plus b y is equal to 0. So, look at this equation, this equation is a second order constant coefficient differential equation. So, we start up with the Euler-Corsi variable coefficient differential equation, the transformation reduce this into a constant coefficient differential equation of course, is second order and this equation can be solved. Now, we are taking the characteristic root, so it is a constant coefficient differential equation. So, this is a constant coefficient differential equation. So, the method is characteristic roots, find the characteristic roots and these three situations where the roots are real and distinct, the roots are real and equal and roots are complex. So, which we have already seen, say for example, if we take the characteristic roots, the roots are given by lambda 1 lambda 2 is equal to 1 minus a plus or minus square root of a minus 1 the whole square minus 4 b all divided by 2. So, the discriminant of the equation b square minus 4 a c by 2. So, if the roots are real and distinct, so call it then lambda 1 lambda 2 are the real roots. Then the solution is given by, solution is given by Y of, remember the independent variable is now changed to Z, Y of Z is given by c 1 e to the power lambda 1 Z plus c 2 e to the power lambda 2 Z. And again, so if I want to change it back in terms of x to change the independent variable back to x, use the transformation. So, therefore, Y of x is equal to you change Z to x that is c 1 e to the power lambda 1 and Z is ln of x. So, ln of x plus c 2 e to the power lambda 2 ln of x and this is nothing but ln of e to the power lambda 1 ln of x is, so c 1 if you simplify it is x to the power lambda 1 plus c 2 x to the power lambda 2. So, this is a general solution. So, general solution if the roots are real and distinct and similarly, if the roots are real and repeated and the roots are complex can be treated similarly. This is the situation where if the differential equation is having a very special form and in case now we are going to deal with differential equation which is in a general form that is a 0 x y double prime plus a 1 x y prime plus a 2 x y is equal to 0. If this cannot be reduced into a constant coefficient equation and no other methods are available to solve it, then one method of solving is by using the power series solution, the series solution of this differential equation. So, let us see how to solve this by using a power series solution. So, power series solution to the equation a 0 x y double prime plus a 1 x y prime plus a 2 x y is equal to and what we expect is, we expect a solution to this equation called it 1, 1 in the form y of x is equal to summation n goes from 0 to infinity c n x minus x 0 to power n where c 0, c 1, c 2 etcetera are constants. So, we expect a solution to the variable coefficient differential equation 1 in a power series form an infinite power series about a point x 0. Now, the question is, does there exist a power series solution to 1 in the form 2? So, called this form as 2. So, does there exist a power series solution to the above differential equation in the form 2 and if it exists, how to compute the constant c 0, c 1, c 2? So, if s, how to compute c 0, c 1, c 2 etcetera in 2. So, these are the two questions. So, first we address the existence of power series solution to 1. So, under what condition the power series solution for 1 exists and that series is convergent? To answer this question, we need some of the basics ideas, some basic definitions. So, to address the existence problem, there is under what condition one has a power series solution of the form 2. So, we require some conditions and we define and let us again state the form of the equation a 0 x y double prime plus a 1 x y prime plus a 2 x y is equal to 0 and equivalent normal form is equivalent normal form. So, we call this equation 1 and if we take the equivalent normal form y double prime plus p 1 x prime plus p 2 x y is equal to 0, where divide throughout by a 0. So, where p 1 x is equal to a 1 x y a 0 x and p 2 x is p 2 x y a 0 x. So, now, we define what do you mean by a analytic function, a function f is said to be analytic, a real valued function f is said to be analytic at a point x 0, if it is Taylor series expansion about x 0 given by summation n cos from 0 to infinity, the nth derivative of f evaluated at x 0 divided by n factorial into x minus x 0 to the power n. The Taylor series about x 0 exist and converges and converges to f x for all x in some neighborhood of x 0 containing x 0. So, neighborhood in some neighborhood or in this case is some interval containing x 0. So, in this case, we say that a function is analytic. Examples see all polynomial functions are analytic, all polynomial functions are analytic everywhere and functions containing e to the power x sin x cosine x, they are also analytic and another important class of functions analytic is all rational functions. So, rational functions of the form p x by q x, where p x is a polynomial, q x is a polynomial. So, rational functions are also analytic at all point except the denominator, the points at which are denominators not 0, except at the point of denominators equal to 0. So, rational functions is analytic except at those values of x at which the denominator is 0. For example, if we take a rational function 1 by x square minus 3 x plus 2, which is written in the form x minus 1 into x minus 2. So, this is analytic at all points except at x is equal to 1 and x is equal to 2. At these two points x is equal to 1 and x is equal to 2, the rational function is not analytic elsewhere it is analytic. Now, a point is another definition is ordinary point for a differential equation. So, definition the point x 0 is called an ordinary point of the differential equation 1. If x is equal to 1, if differential equation 1 is in the normalized form, call it y double prime plus p 1 x y prime plus p 2 x y is equal to 0. If both functions p 1 x and p 2 x are analytic at x 0, a point x 0 is said to be an ordinary point of the differential equation in the normal form y double prime plus p 1 x y prime plus p 2 x y. If both of the functions p 1 x and p 2 x are analytic at x 0, if that fails, if it is not true then we say the point is a singular point. A singular point if either or both of these functions not analytic at x 0, then so then x 0 is called a singular point of the differential equation. See quickly let us look into an example y double prime plus x y prime plus x square plus 2 y is equal to 0. So, here p 1 x is just x and p 2 x is x square plus 2. So, obviously both are analytic for all points, all points are ordinary points for this differential equation. All points are ordinary points and if we consider another differential equation x minus 1 y double prime plus x y prime plus 1 upon x y is equal to 0. So, here p 1 x is x upon x minus 1 and p 2 x is 1 by x into x minus 1. So, p 1 is analytic for all points except at 1. So, for p 1 x is equal to 1 is a point at which p 1 is not analytic. And here x is equal to 0 and x is equal to 1 for p 2 is not analytic at these two points. So, therefore, the differential equation is analytic for all points except 1 and 0. So, the conclusion is 0 and 1 are the only singular points. So, this tells us x is equal to 0 and x is equal to 1 are singular points of the differential equation. Now, if x 0 is an ordinary point of a differential equation, then we have a sufficient condition to guarantee a power series solution. So, now I state the form of a theorem. Theorem says if x 0 is an ordinary point is an ordinary point of the differential equation, call it y double prime plus p 1 x y prime plus p 2 x y is equal to 0. Then it has two linearly independent non-trivial two linearly independent power series solution of the form y x is equal to summation c n x minus x 0 to the power n, n goes from 0 to infinity and the series converges in some interval, the interval of convergence x minus x 0 less than some r. So, this is an important theorem guarantees the existence of a power series solution to a differential equation. If x 0 is an ordinary point of a differential equation, then about that point we can find, if you have a second order equation, then we can find two linearly independent power series solution. So, existence of two linearly independent power series solution at an ordinary point is guaranteed by this theorem. So, now the method of solution, how to compute the solution? So, method of solution is, by method of solution, what do we want to find? We want to find to find c 0, c 1, c 2, etcetera all these coefficients in the expression y is equal to our series solution is of the form c 0 plus c 1 x minus x 0 plus c 2 x minus x minus x 0 square plus etcetera, which is in compact form we write as c n x minus x 0 to the power n, n goes from 0 to infinity. So, our aim is to find these coefficients, our aim is to find these coefficients. So, since a series converges on x minus x 0 less than r, some number r by the existence theorem, since a series converges for x minus x 0 is less than r about the ordinary point x 0, the series can be differentiated term by term, it may be differentiated term by term on this interval. C d y by d x is derivative of y, which is c 1 plus 2 c 2 x minus x 0 plus 3 c 3 x minus x 0 square plus etcetera, which is summation n goes from 1 to infinity n c n x minus x 0 to the power n minus 1. And similarly, second derivative d square y by d x square, the derivative of the about series, which is 2 c 2 plus 6 c 3 into x minus x 0 plus 12 c 4 into x minus x 0 square plus etcetera, which is written as summation n is equal to 2 to infinity n into n minus 1 into c n x minus x 0 to the power n minus 2. So, now, substituting these values of y d y by d x d square y by d x square in the original equation, now substituting y d y by d x d square y by d x square in the differential equation. And simplifying, we get some constant k 0 plus k 1 into x minus x 0 plus k 2 into x minus x 0 square plus etcetera, is equal to 0, where these constants k 0, k 1, k 2, these are functions of our other constant c 1, c 2, etcetera. So, since the series, since the series, it is valid for all x in the interval, all x in the interval x minus x 0 less than r, that is the interval of convergence. And now, equating right hand side and left hand side, the coefficients of x, x square, x cube etcetera and also the constant terms, we get k 0 is equal to 0, k 1 is equal to 0, k 2 is equal to 0 etcetera. So, we get equations k 0 is equal to 0, k 1 0, k 2 0. If you solve these equations, then we get the values of c 0. So, solve these to obtain values of c 0, c 1, c 2, etcetera. And once we have c 0, c 1, c 2, etcetera, plug into the series form, we get the series solution. The series solution y x is equal to summation c n x minus x 0 to the power n. So, this is the method. So, let us illustrate this by an example. So, let us consider an example. So, what is the problem? The problem is the question is to find the power series from the power series solution of the differential equation, find the power series solution of the differential equation given by y double prime plus x y prime plus x square plus 2 y is equal to 0. So, find the power series solution of the differential equation in powers of… So, the question is to expand or get the power series solution in powers of x. So, that is you are asked to expand, get the solution in powers of x minus x 0, where x 0 is 0. Let us check whether 0 is an ordinary point.  So, all points are ordinary points. So, all points are ordinary points. So, x 0 is equal to 0 is an ordinary point. So, therefore, by the existence theorem, there exists a power series solution. So, two linearly independent power series solution of the form summation c n x x x to the power n. So, we look for a solution y x, y is equal to summation c n x to the power n, n goes from 0 to infinity. So, this is guaranteed by the existence theorem. So, there are two linearly independent power series solutions. So, what we want to do now? Our aim is to find this c 0, c 1, c 2, c n. So, what we do is by differentiating, differentiating we get y prime. y prime is summation n goes from 1 to infinity, n into c n x to the power n minus 1. And similarly, y double prime is summation is going from n is equal to 2 to infinity, n into n minus 1, x to the power n minus 2. Now, plug in these values to the given differential equation. So, substituting in the differential equation, we get summation n into n minus 1 c n x to the power n minus 2, the first term that n goes from 2 to infinity, n goes from 2 to infinity, the first term. And the second term is x into y prime, which is x into summation n goes from 1 to infinity, n into c n x to the power n minus 1. Now, the third term is the sum of two terms, x square plus 2. I take x square into summation into y, summation c n x to the power n, n goes from 0 to infinity plus constant m 2, 2 into summation n goes from 0 to infinity, c n x to the power n, which is equal to 0. Since x is independent of the index, we may rewrite this. So, rewriting, we get summation n goes from 2 to infinity, n into n minus 1, c n x to the power n minus 2, the first term, plus summation n is equal to 1 to infinity, n into c n x to the power n plus summation, if we have just multiplied each term by x, we get x to the power n and n goes from 0 to infinity, the third term. There also, we did a multiplication by x square. So, that results c n into x to the power n plus 2, when multiplied by x square plus 2 times summation n goes from 0 to infinity, c n x n is equal to 0. Now, look at the first and third term, this first term and the third term. The first term, the index is n minus 2, the third term, the index is n plus 2. We want to make all the same uniform index that x n. So, we can do the following to rearrange that. So, consider the first term. So, consider the first term, first term, first summation and replace n minus 2 by a new variable, new dummy index m. Therefore, what we have is m is equal to n minus 2 or n is equal to m plus 2. So, if you use this, then the summation becomes, then this gives the first term in terms of m. m goes from 0 to infinity. Now, n is going from 2 to infinity, that becomes when n is 2, m is 0. So, the summation goes from 0 to infinity, m plus 2 into m plus 1.   We can change m to n, change m to n notation. Therefore, it becomes summation n is equal to 0 to infinity, n plus 2 into n plus 1 into c, n plus 2, x to the power n. Similarly, if we do the same thing for the third term, then we will make n plus 2 is equal to m or n is m minus 2. So, this gives in terms of m, this will be m is going from 2 to infinity instead of n is going from 0 to infinity and m is going from 2 to infinity, c m minus 2 and x to the power m. Again, that is m is a dummy index. Therefore, we can plug back the n. This is n goes from 2 to infinity, c n minus 2, x to the power n. Now, the index of all these terms are n with respect to n. So, what we do is the equation now becomes summation n goes from 0 to infinity, n plus 2 into n plus 1, c n plus 2, x to the power n plus summation n goes from 1 to infinity, n c n. Therefore, we do not do any change, x n plus summation, the third term is also changed, n goes from 2 to infinity, c n minus 2, x n plus the last term, 2 times summation n goes from 0 to infinity, c n, x to the power n is equal to 0. Now, we want to make some common range. Since the summation is not uniform, some one is starting from 2 to infinity, another one is 1 to infinity, another one is 0 to infinity. The common summation range is 2 to infinity. So, the other terms we can separate out. For example, the first term, first summation, the case n is equal to 0 and n is equal to 1 can be separated out. Therefore, what we get is this term, which is 2 c 1, 2 times, when n is equal to 0, get 2 c 2, 2 c 2 plus 6 c 3, x plus the common range, n is equal to 2 to infinity, n plus 2 into n plus 1 into c n plus 2 into x to the power n. Similarly, for the second term, that can be split as c 1 x plus the common range, n is equal to 2 to infinity, n c n, x to the power n. And third term, we do not have to change. The last term, we can change this is 2 c 0 plus 2 c 1 x plus summation c n x to the power n, n goes from 2 to infinity. Now, if we rearrange all this, we get this implies that 2 c 2 plus 6 c 3 plus summation, the first term. So, if we combine all these things together, if we add them together, we get 2 c 0 plus 2 c 2 plus 3 c 1 plus 6 c 3 x plus summation, the common summation, n is equal to 2 to infinity, n plus 2 into n plus 1 c n plus 2 plus n plus 2 c n plus c n minus 2 times x to the power n is equal to 0. And this series converges for in the interval x minus x 0 is less than r. So, equating the coefficients of the powers left hand side to 0, because right hand side is already 0, we get 2 c 0 plus 2 c 2 is 0 and 3 c 1 plus 6 c 3 is 0. And this gives c 2 is minus c 0 and c 3 is minus half c 1. And also, we get from the summation term that n plus 2 n plus 2 into n plus 1 c n plus 2 is equal to plus n plus 2 into c n plus c n minus 2 is equal to 0. If we solve it, c n plus 2 is obtained as minus n plus 2 c n plus c n minus 2 divided by n plus 1 into n plus 2 for n greater than 2. Therefore, for each case, n is equal to 2 case, c 4 can be solved, c 4 is minus 4 c 2 plus c 0 by 12. So, this implies that c 4 is 1 by 4 c 0. And similarly, n is equal to 3 case, c 5 is solved to be 3 by 40 c 1. So, therefore, the solution can be written as, the solution can be written as by using these coefficients. So, y of x is equal to c 0 into 1 minus x square plus 1 by 4 x to the power 4 minus etcetera plus c 1 into x minus half x cube plus 3 by 40 x to the power 5 plus etcetera. So, we see that there are two series. So, this is two series solutions, the first one and the second one. They are linearly dependent, two series solutions and their linear combination c 0 of the first one plus c 1 of the second one is a general solution, say general series solution. So, therefore, by doing this method, we can find, if a point is an ordinary point, we can get the series solution and two linearly independent series solution of a differential equation, if the point is an ordinary point. And if the point is singular point, there are methods, Frobenius methods and all that will come in another series. So, with this, I would like to finish. So, we have seen, even if the differential equation is variable coefficient, we can have a solution in the in power series form, series solution. Bye.