I am ready. I feel the Holy Spirit in this place. I I love it to be I love when Holy Spirit come comes I Believe I truly believe God has a purpose for this service I am waiting to the end But sometimes he show up and he said enough enough my son I Want to serve I want to touch my people Hallelujah, I'm excited to be with you Yes, Pastor Eli will go around the world we will be in every nation Even though we already been in more than 30 countries and Every month we are in a different place. We are flying to Russia in two weeks August will be in Russia September will be in Germany October. We're coming to Nigeria for the glory night in November we will do in crusade in Ethiopia December will be in other part of Ethiopia doing crusades. Oh God is good He's in a good mood He's doing good things and this is the good news. Come on somebody God is good Maybe it will be revelation for you, but let me tell you something God always in a good mood He always in a mood to heal people deliver people set people's free safe people and This is the good news of goodness of God and his greatness His mighty name. Amen recently we came from Ethiopia and And in Dubai, but in Ethiopia, I mean I have so much testimony so much stories to share How power of God set people's free and deliver people and heal them but one of the one of the Woman just catch my attention One of the lady was died with chains for 20 years 20 years her legs Arms she was tied with chains and ropes for 20 years Somebody brought that girl to our crusade During the service power of God just hit that girl In a mighty way we prayed God deliver her by the end of this service We told her parents to untie her they're like no way We cannot do this because you don't know what's going to happen. We see we see what we told them watch this untied her When they untied her she got totally delivered She starts singing worship songs At the stage and she said that when she was a teenage teenager She was in a choir singing the worship songs And she said I've been always want to be a worshipper I've been always want to be a sin sin to God So when devil, you know just tied her with with bandage and evil spirits for 20 years My eyes is a witness a front of everyone God deliver her and she star worshipping Jesus the mighty king. I Told the story this morning. We came from Dubai. It's Islamic countries a country and during our cross Conference one of the Larry came from one of the Islamic countries. I don't know there are seven of them So she flew from different country. She never heard about Jesus What happened with her in her city bump was explored and half of her body was numb So she opened Facebook and she saw advertisement of healing deliverance conference So she flew from her country to that conference During the service the Holy Spirit touch her and heal her completely So she start hearing seen the body become normal during the service. Nobody pray for her No, but I did not pray for her I was preaching gospel and I was preaching about the power of God that the God himself touch her heal her Restore her she came after the service to the pastor and she said I want to have that Jesus that you guys are preaching This is what I'm talking about when God show up when the power comes Come on everything bows That's what we are desperate to see And this is what I believe That the church in the last days will be glorious full of spirit and power and fire That people that was my desire from the beginning when I When I gave my life to Christ, I I told Jesus. This is my true story. This is my testimony My wife is a witness. I'm glad she's with me with my daughter here today I I told God I said I don't want to be normal Christian.  I said God. I don't want to be normal. No, this is not This is not the price that you pay for my soul. I Believe everything in this book everything today Everything what you're saying in the book of eggs. I want to see in my life That's my prayer. I was a God take me all I want to be available for you Whatever you want from me transform me change me deliver me set me free Equip me send me but I want to see this book alive in my soul in my body In my mind. I want to see everything what you wrote in this book. That's my prayer request And let me tell you it's still happening tell this not tell this day That's a process That's a process of deliverance Hallelujah Say God I am ready. I Want to be available for you Yes father, we give you praise and glory in this place right now Manifest your power your glory And we give you the praise Holy spirit, this is your platform and I'm yours use me speak through me equip these people set us free Bring us to the knowledge Let the spirit of wisdom and revelation rest upon us this day We receive this by your grace and We praise your name and only your name about all names in Jesus Christ We pray a man and a man and a man. Oh Expect something is gonna happen I Want to I want to read John chapter 1 from the message translation if you have a Bible Please open with me John chapter 1 I Love second service because The biggest Problem I have I'm struggling when I'm limited When people Telling me you have only 30 minutes or 40 minutes. That's it.  So I'm not I'm not there yet But I'm in the process You know So that's why I love second service because there is so much more in me to release and I want to release I Want to release and I believe you will be blessed God is about to do something John chapter 1 Verse 6 if you dare with me from message translation, I want to read 1 6 Follow me, please There once was a man his name John Sent by God To point out the way to the life light Here this he came to show everyone say everyone He came to show everyone where to look Second who to believe in John was not himself the light he was there to show He was there to show he was there to show he was there He was there that would catch my attention. He was there. He was there. It's a place Where God plays him he was there. Can you say with me? I want to be there Yes, I want to be there He was there right place right time right message Right place right time Right message. I want to be there. Come on somebody. That's your purpose That's your purpose. I want to be there why to show To show the way to the light I Want to be there to show the way To the light basically he was saying To help people to show people have to get there To show people have to get there So Bible says there once was a man his name is John sent by God That was my prayer from the beginning when I gave my life to Christ in 2002 16 years ago I pray to God and I said God. I want to be sent by you. I Got this revelation from the beginning. I believe many people they They they they start the ministry in a wrong time with the wrong character So sometimes they are in the right place, but wrong time So we need to be Sensitive to that moment. We really need to wait upon the Lord.  You will have a noixin. You will have power You will have fire. You will have seal. You will have passion No one can stop you can stop you No circumstances nobody When you know that God is sending you you have to understand it's a process Process or what watch this the first thing what Bible says about the John the John The Bible says that he Came to show watch this he came to show Not himself He came to show not his talent He came to show not his gift Who he is have he can preach have he can minister have he can heal? That's not the main point Bible says that he was there to show Everyone where to look Not on himself That's maturity. That's character that that's that's that's how you know who you are in Christ When I live no more and he lives through me To show where to look not on himself, you know many times We do something for God we preach we teach we sing we whatever we do But deep in our heart Deep in our heart will really want wants people to see us like have good Talented I am have gift a gift that I am you know like watch how can I have how can I preach watch? How can I do sermons how watch how? How much wisdom I have or knowledge of I have let me tell you something the biggest thing that you can do It's just to be available for the Holy Spirit just to be empty For the Holy Ghost to use you to use your body to use your mind Damn your skills your gifts your talents will have a purpose and meaning and values So best thing what we can do through that process it's allowed God to deliver us That's the process to show everyone who to look Not to ourselves, you know when I gave my life to Christ I you know what I what what I start I start with a drummer I'm a musician myself. I play by the way. I play every instrument every instrument I'm writing albums right now and I write songs and music and I produce the music. I play all the instrument That's my gift. By the way, I'm writing the book too. So Is gonna come out by the end of this year is gonna call predestined You need to get that book It's about my life story with a revelation and the renewing of the mind and to know your purpose in the will of God Here on earth as it is in heaven. So this is just small commercial So when I gave my life to Christ, you know, I start is a drummer nothing wrong. You you're the best drummer I ever seen This guy just amazing But I was playing, you know, a little bit just And and and and that there was a small group, you know, just the guitar keyboard man and drums Nothing else. It was a small church and and Can you imagine? I was there the leader of the group before and When I gave my life to Christ God put me in inside of that thing, you know just observe and You know, not to have attention to my Myself just watch people and play and there was a no chin woman of God was leading the worship But she got pregnant and she needs to live and then pasta have a two choices One of the choice it was me and another choice was somebody else So he was choosing between who's gonna be the next worship leader And you know, I thought come on I can't do anything here at the platform. I can write song singing and lead I grew up at the stage Okay, all my life. So I was like, come on. I can do better than this person and I thought pastor would choose me But this is what happened that another person called me and Said I need to come to your house. I Said come on come to my house. So when that person came to my house, I don't want to mention the name Because maybe that person will watch this It's a blessed blessed person Anointed by anointed by God by God so That person came to my home and said to me, you know, I was praying and God spoke to me That I will be the next worship leader.  I was spending time with the Holy Spirit all night and the end of the night and in the end of there, you know, my Devotion angels came to me Gabriel with a message and he told me no son You will be the next worship leader. I Didn't say that Bless said be the name of the Lord. I didn't say that but I was about to say that So I said, okay If truly God spoke to you That you will be the greatest next worship leader Fine do it. I Will continue to play drums But when I came to my room at night to pray because I was waking up three o'clock in the morning every single night to seek the Lord Holy Spirit spoke to me and he said son, it's not about gift It's about character if you will know what I'm preparing you for I Don't need another musician. I Need the man with vision in Mission with purpose that he will use Whatever gift he has with the right attitude inheritance Integrity it's about attitude your character Then Holy Spirit told me he said now.  He said then watch what you have to do now all the ideas That you have have to build the worship team. I Want you to help that person? To do exactly the way you wanted to do in the team Don't you ever ignore the process That's the most important moment in your life Cause he doesn't want another man who know how to preach He wants you to be there with the right attitude right moment right place Right message not just to carry the message that but to become a messenger To have a message That your life will present that message that through your life you can show Where to look who to believe in and have to get there Never forget one time Before I start international ministry I start started in 2010, but I gave my life in 2002 and For seven years seven years my wife is a witness for seven years I am I continued. I'm continuing to do this by now, but there then I had more time Because I did not travel so much So I was serving the local church and I become one of the pastors in the local church, but I was rejoin myself every single month every single month To the mountains and Spend time with God for three days every single month sometimes we didn't have any money at all I borrow money to pay for that room just to go and be with God And it was happened every single month Months to months months to month and I remember one night. I was there seeking the Lord Dwelling in his presence in the middle of the night when the Holy Spirit just Came to my room and I saw a vision and he said to me he said I want you to see your end From this moment. I want you to see your funeral.  We're never gonna die It's a transition from glory to glory From life to life Don't worry about the day that they will come anyways Soon or so later So we need to be prepared for that day Are you with me? So golden spirit spoke to me. He said I want you to see your day of your funeral What do you want and I mentioned three things This is the true story.  I'm not gonna say this to you because he will be jealous. So I Saw that day I start start like imagine that day have that image inside of me and first thing I said to God I said number one what I want That I'm my funeral nobody will cry This is what I want. Ah, not not this not this hold on Number number one.  I said to him. I want to preach my last message. I Want to sit down on my chair? Wave to everyone and say I see you there and go I Said that's my request. I want to preach my last message. I don't want to die in a car accident, you know Through I don't know somewhere in the air with airplane crash. No, no, no, no.   I want everyone. I'm Telling this now that camera will record so after the hundred years I don't know if we'll still have the same cameras, whatever, but but I want them to remember remember Not my enemies, but their children Remember that they need to wear white clothes on my funeral That was my three requests to Jesus But then he shut my mouse. This is the true story. The presence of God was so heavy in my room and he said You want to see this I said yes, he said then let me tell you have to get there Do you want to know? He said When person is dying what he is thinking what he is saying, let me tell you when you or somebody is dying They will never think what house they lived What kind of car they drove? How much money they had They will think Have they use the time have they use the house that they live Have they use the money that they had have they use The that the gifts He he said to me. I want you to think right now The most valued things in life There is a principle in life principles He said I want you to think about those principles Like this is your last day on earth now he said get those principles and Now come back to your time where you are now and Put those principles in front of you now and Those principles will lead you and will take you there and When you're gonna get there you will protect your heart and your mind and your soul and your ministry in your kids life And he said you will see that end that you require It's a process It's a process and We need to go through that process That process Will help you To have that belief system inside of you. I Said God, what do you want me to do? He said look Ezekiel chapter 3. I want to read quickly Ezekiel chapter 3 He said To the Son of Man Eat what you see Eat this book See Through that process you have a choice through that process God give you choice to make What you put in front of you what you're gonna see and what you're gonna eat? That's a responsibility That's a responsibility that will help you to grow In your character, I believe most of you You love people who's responsible Self-control discipline What is that? It's a fruit of your character Through the process I Don't have time to share my story and all my process. I just want to give you some principles But God says to Ezekiel.   So I ate it.  He said I want you to take responsibility and I want you every day To dwell in my word until my word will become flesh in you And my word will reveal the truth and the truth will set you free and the truth When set you free will make you available for me to have a message See, it's it's not about how what you speak or what you say It's about the character that caring that message without compromise without compromise when you're alive when you stand firm when you are very strong in your values and principles and and Character and and the more you dwell in the word the word will reveal you who you are in him And what is your name because when you go preach?  I never Knew in my life that I will be in love with this book by the way, you know in the beginning when I Start seeking the Lord and thinking about Worldwide Ministry God told me he said I will teach you to serve one person and And you want to see, you know thousands and millions And by the way, we work in to have a 1 million souls crusade In Ethiopia, but before you see those massive Crusades and you know worldwide ministry He said to me I will teach you to see value of one soul I will teach you small things have to be faithful. I do remember I said Natasha, we're gonna go around the world Because Jesus revealed to me Will be in every nations to preach the gospel of the kingdom and she said to me, you know what Andre? I think you should read at least one time Bible from the beginning to the end and let me tell you something I Opened Genesis. I opened the Bible 16 years ago And God is my witness. I never close this book Tell this day and this book place me in a position where I am now Through the process I remember one time I was flying from Kiev to New York and Sitting at the airplane somebody before I left gave me a book called Man that got used So I opened that book start reading that book When airplane was about to take off One of the lady was sitting across an aisle seats. I love aisle seats off sitting, you know an aisle seats and then this Have you called? aisles It was a crossing me one of one of the Russian lady and When the airplane was taking off she started dying Like coughing screaming shaking, you know, like she was doing like this and I Look at her because I was reading the book man that got used, you know, and when I look at her She's dying there And I decided I'll pray in the Holy Spirit and I start praying like God, please touch this woman You know, please and God spoke to me. He said I am the spirit. I need your body So lay hands on her and I want you to pray for her.  He's like come on Do you believe what you pray and you need to believe in him? That's a belief says that your mind said He said if you believe I can do this then lay hands and be available for me I want to use you to heal this person So I pray in the Holy Ghost, you know the shabra also to a Holy Spirit touch this woman and the more I pray the more I hear his voice stop praying Look at this woman lay hands and pray for her But is any of you flights, you know, you ever flew in a airplane, you know when everybody's quiet And the plane is taking off, you know, and everybody just watching you know, look it from the windows And so quiet and then can you imagine this like God is saying to you come on pray and cause the demon out of that woman But the scary part I was holding that book man that God use So I close that book put it away. It's the pain of the ghost Holy Spirit touch this woman and a third time God just you know He said Look at this woman lay hands and pray and command that spirit of death to go So when you overcome the fear Anointing comes Let me tell you from my own experience I can tell you story-to-stories in Pakistan and Congo in different places where God he was helping me to overcome The fear and then heaven was open and God brought his spirit when I overcome the fear. I Felt boldness Like I see millions people in front of me at the crusades, you know, this is just one little woman sitting there But when I was praying for her I felt like I'm praying for thousands With boldness and anointing and by the way when I lay hands on her some one of the girl was walking through the aisle You know, and I was like wait over there. I'm praying right now Seriously, I scream like this because I cannot speak normal. I have to you know, just shout and I said Much. Oh, no, no, that's not that I said listen to this I don't have time to explain to you who am I who I am. I say it's not about me It's about the one who I'm present right now who to look for you know what I'm saying It's about the one who to look who to believe in In the place where I am right now to show the way come on somebody in the place What I am right now to show the way I was in that place And and and I said to this woman I said I want to pray for you may I pray for you?  She said I cannot I cannot fly Because of my heart problem. I have metal stick metal thing metal clap and whatever in my heart Inside of my heart. She said I signed the paper doctor paper That all my responsibility to fly to New York on my own responsibility Because she said I want to see my daughter. I didn't see her for 10 years So I signed the paper if something is going to happen with me. I'm responsible for that So she said when the airplane was taken off I start my heart stops and she said I lost my Hearing and I couldn't speak and I felt I she said I seriously felt like I'm dying and She said to me. I thank God that you was there come on somebody said that you were There that you were there in the right place in the right time with the right message Come on. You were there with the right place with the right message She said I give praise to God for your life For your life for your appearance for your appearance I give praise Hallelujah Come on. We cannot ignorant that process What is that process just few things few thoughts and we're gonna pray When you there in the right time with the right message See the bible says Those who faithful in the small things see we're always ignorant those small things Because we think That what could happen through those small things it doesn't matter what what this thing has to do Has something to do with my purpose with my calling That's the way we think But you must understand that God Desire your purpose more than you does the reason you exist on earth So we always tell God God. I want to know my purpose. He said back to you You have to understand that's the reason why I send you to earth It's not about purpose. It's about character That will help you fulfill that purpose Are you with me? Because gift will lift you up, but only character will keep you there So it's not about the purpose It's about character. Have you present the message? Have you live the life because your life will show the way? Not your words And you have to understand message bigger than your words You got to use your words to express To reveal the message but message. It's your purpose. It's your life. It's your mindset.   So we think that god is watching us when we're here at the stage You know grab the microphone and and and and and and we'll know how to pray and praise and lift our hands And shout and scream and we'll know how to preach and release the word of god No, he's watching you in those two things that matter for him It's faithfulness because faithfulness is the power of your discipline Faithfulness is the power of self-control This faithfulness isn't the part of your character If you faithful in small things believe me that they will come You will be in the right place in the right time and god will use the mighty way and he will be make sure He can trust you Believe me. I met many young people. They know me from the beginning They're watching our ministry right now and they want to be in our place in our position Escaping the process See when your church is praying for the thousands locally and and and millions globally I'm I'm telling you it has something to do with this building with cleaning with cameras with faithfulness in the small things When no one sees you When everybody is eating at a restaurant, you know seafood and shrimps and lobsters And you buy yourself in this building. You are cleaning this carpet This is where god is building your ministry your purpose in jesus name And let me tell you The anointing is growing there The power is growing there The purpose is growing there You grow all in that mission You know you grow in that purpose in that character you become that believing system inside of you And when you're faithful in others When you start focusing in those two things Those two things you can fulfill when you continuously seeking his face Through his word because Him himself will give you strength and passion And let me tell you you need more strength in the small things Than strength for the big things Because in the small things You hope you have to overcome your desires You need to have a power over yourself This is so powerful. Let me tell you something in the small things You need to have a power overcome your own desires So you need more strength from god more passion from god more zeal from god With the small things Then when everybody knows you watching you reading your books and your flow Because people is helping you But with the small things you need god strength to do it not people's strength You know what i'm saying you need to see that god is on your side is helping you and watching you in those small things And he's watching careful how you do ministry in someone else I truly bless by this ministry. I'm telling you with all of my heart. I love hundred generation I believe god has a big purpose for this local church this local church more than this waltz This local church. It's a gifts purposes You know the the lives of these people who will think globally and manifest his kingdom in every corner of this earth So I truly believe in this ministry And i'm glad to be part of friendship with these leaders and pastors and let me tell you something If you really want to grow you need to have that right environment You need to have that soil where you can grow and i'm prophesied to you I'm declaring the word you need to plug your life in the soil in this environment And start with the small things Start with the small things and believe me those small things Will take you to the big things to the higher level and the purpose of jesus christ in your life small things and uh ministry and others You know churches and and businesses whatever god place you so and the end i want to mention this Where you are right now That's exactly the place with god will god place you there This is your place now So my question to you What do you see from that place? How do you do things from that place? What comes out from you from that place? And what kind of message you caring from that god i deserve more Oh, I don't have time I don't have time Another time I deserve more more They don't see my potential They don't know how I can do things Where you are That's the place will brought you through the self control and discipline in your own time and life Let me tell you that that where you are right now.   There is a place For your destiny There is a place where god will use you in a mighty way To show the ways To show everyone who to look and who to believe in Father i give you praise right now in this room. I give you praise right now in this service I give you praise and jesus We magnify your name About all names If it's possible everyone in this room just open your mouth and pray the holy spirit pray loud Release the prayer Release the spiritual energy From inside to out In the mighty name of jesus. Thank you for watching this content I hope this was a blessing to you if you're like me and you like to click on things Click on this subscribe to our channel and the content will come to you every time we post it and remember The best is yet to come


This weekend, we feature our own senior fellow, Mark Thornton, in an appearance on Paul Malloy's FreedomWorks radio show. Mark is known for his work on the Skyscraper Index model, which can help us understand why booms are followed by busts, why malinvestment is inevitable in an era of artificial interest rates, and how central banks cause so much harm and misery. Building booms, especially in the context of big city skyscrapers, can be clear signs of dangerous bubbles and hubris. It's no coincidence that the Empire State Building was built on the cusp of the Great Depression, and Moscow's new Empty Financial Center is an eerie reminder that phony growth can end very quickly. With huge megatowers planned for China, Korea, Saudi Arabia, United Arab Emirates, and others, the Skyscraper Curse may strike again. We hope you enjoy Mark's great interview with Paul Malloy. Stay tuned. We're delighted to have our old buddy Mark Thornton join us from the terrific Lidwig von Mises Institute in beautiful downtown Auburn, Alabama. And he is brought to my attention something I have never heard of. It's called the Skyscraper Curse. And apparently it's something that's been around for a while. Mark, welcome back and tell me what in heaven's name is the Skyscraper Curse? Hey Paul, it's great to be back out of the air with you. The Skyscraper Curse is something that came to my attention about 15 years ago. And it's basically the unusual correlation between the building of the world's tallest skyscraper and world economic crisis. And I know that seems really farfetched, but I can explain why there it really is a correlation here. And it dates all the way back to 100 years ago with the panic of 1907. And at that time the Singer building was constructed making it the world's tallest skyscraper. And it opened its doors in the aftermath of the panic of 1907. And we've seen this correlation appear over and over again at the height of the biggest bubbles over the last century of time. You know, it's interesting you mentioned the Singer building. I remember from history that the Empire State Building was built in 1930, completed in 1930, one year after the stock market, the great stock market crash. Yes, that's right. There was several large buildings built in New York City, the epicenter of the stock market bubble in the late 20s and early 30s, including the Chrysler building, which set a record prior to the Empire State Building. But the Empire State Building set the record during its construction, almost the same time that the stock market peaked in late 1929. And then of course, there was a tremendous meltdown in the US economy and the world economy really. The America's Great Depression was a worldwide Great Depression. And so when the record is worldwide for the world's tallest building, usually the economic crisis is also worldwide. And of course, the Empire State Building way ahead of its time in terms of the mammoth record. And it held that record until the early 1970s. Right. Now, a lot of people would say this is a non sequitur. There is no correlation between building a tall skyscraper and an economic collapse or an economic crisis. And you would say what? Well, I know there's a lot of indicators out there that are silly, like who wins Superbowl determines how the stock market goes. There's a lot of those silly, just various correlations out there. But there's a reality here behind this connection, behind this correlation because to get a world economic crisis, you have to have a long period of sustained artificially low interest rate, which invigorates the economy and it gets people to start all kinds of construction and all kinds of investment because of those low interest rates.  And so if that setting of artificially low interest rates, a boom in the economy, lots of speculation, new investments without track records is when these types of buildings, these types of projects are undertaken. And so there's a precise economic reason why record setting skyscrapers are built. And those same reasons are what causes bubbles in the economy, which inevitably have to bust and cause recessions or depression. Now that is a very, I mean, now as I think about that, that makes a great deal of sense. By the way, we're talking with Mark Thornton of the Great Ludwig von Mises Institute, an economics, Austrian economic think tank out of Auburn, Alabama. And Mark has been with us before many times talking about different things, but this one is really interesting.  In 2014, I think we may have set a record for some of these skyscrapers that were built. Yes, that's correct. Last year, there was the record number of skyscrapers in process of being built and completed. There was 97 of these buildings that exceeded 200 meters, which is 650, six feet high. That's the minimum. So you got 97 of those. And then you have, which beat the old record by a tremendous amount. And the number of skyscrapers in existence is now approaching 1000 of these whopping tall buildings. And so the number that have been built over the last 10 to 15 years has been enormous.  You go to China, especially that's where most of the beats are taking place. China is set to open its tallest building, the Shanghai Tower, in a few months. And that would be the third tallest building in the world. And also in Asia and also in the Middle East, they've built a number of super tall buildings in Moscow. And the Shard building in London, which opened a few years ago, set the European record and then ushered in the European economic crisis. And so this, you know, the building of super tall buildings has really picked up its pace. And now in Saudi Arabia, they're set to build a building that is one kilometer high. I don't know how many stories it's going to be, but it's going to easily surpass the old record that was occurred in Dubai in mid 2007. And of course, that ushered in the housing crash and the worldwide financial crisis. Wow. Okay. Well, let's go back to that. Now when Dubai built that skyscraper, the Burj Khalifa, I believe it is, and that was back in 2007. So that was about a little less than eight years ago, maybe. Did something happen after that or around that time? Oh, yes. It set the record on July 21, 2007, which was the top in the housing market. And if you go back and look at the Philly housing index, that's when it peaked, was in July and August of 2007. And as a matter of fact, aside from the skyscraper thing, I have a blog post on lurockwell.com in August of 2007, showing the Philly housing index topping out and starting to crash. And so this, the skyscraper curse indicator had me glued into the situation. Really? You were actually on the lookout when you, you were on the lookout for something like this back then? Oh, yes. I had about the housing bubble in 2004, 2005, and 2006. And I've been doing speaking arrangements, engagements, excuse me, during that period of time. And I was a very disliked person for what I had to say back then. Were they throwing tomatoes and stuff at you, as you told them, that this kind of thing is indicative of economic trouble on the horizon? Yeah, I didn't give any talks during tomato season at all. But I had some angry audiences. I can tell you that for sure. Oh, yes. Well, Mark, so we have all of these skyscrapers built last year. Does that mean we're in for a depression?  You know, the more skyscrapers, the worse things they're going to be. I mean, so they've been borrowing, so these obviously, let's go to this 2014, all these people building these big buildings because there's easy money, right? There's easy money out there. And so everything's going to kind of come down to a crashing halt here this year.  Well, the skyscraper index curse indicates that the Chinese economy should be contracting in a significant way this year. And then next year, late next year, the project in Saudi Arabia should be about setting the new record mid 2016. I believe if the construction stays on track, which it doesn't necessarily, that would indicate a worldwide collapse. But the Shanghai tower in Shanghai, which would be the Chinese record, is an indicator that China should be having significant economic problems. And of course, I don't have any eyes on the ground in China. And the statistics are kind of difficult to interpret because they're all doneied up by the government and the state banks. But the statistics that I have seen are not indicating a positive economy in China. It's contracting, growth is missing its target, and the price of oil, which the Chinese economy had been stimulating, of course, has come down and is projected by me to come down significantly more, which is good news for your audience, but not necessarily a good indicator for the Chinese economy or the worldwide economy. And I don't know if you've heard of this, but there's an index of shipping cost called the Baltic Dry Shipping Index. And that's an indicator of the price of how much it costs to ship goods across the oceans and through the Panama Canal and through the Suez Canal. And that index has been as high as $11,500. And right now, it's sitting at about $525. Really? So that indicates to me that while the amount of ships in existence has increased, that the demand for those ships is shrinking significantly. And so that's an indicator that the worldwide economy is contracting at this point, but not necessarily crashing at this point. Wow. Now, these ships are not, obviously, there's not as much cargo as what you're saying, being transported. Is it mostly oil that has gone down, or what? Well, no, it's not oil, because this is the Baltic Dry Shipping Index. So this is the type of ships where they put those big containers on. Okay. And so it's all things that are in containers. And it includes things like coal, it includes things like iron ore, things that the Chinese economy had been importing vast amounts of. And so this is an indicator that there's less economic activity in the world, and that there's less trade in the world, and that the countries that export and the countries that import are having a difficult time. And I can tell you that, for example, the Brazilian economy, which exports a lot of minerals and sugar and things of that nature, that's in a contraction for sure. The Russian economy is in a contraction for sure. The Chinese economy is missing its targets and contracting. And so these are big players in the world economy, and players that were accounting for a lot of economic growth in the world economy. And so in fact, the US economy is not contracting, but expanding at a very disappointing level. And of course, the latest GDP reports will revise downward. So I'm very concerned about the worldwide economy at this point. Yes, the stock market's near and all time high. I mean, all of the indicators that you're kind of talking about, kind of like what happened back in 1929, too, just before the crash. And, you know, the artificial levels and interest rates are low artificially. So are you predicting from all of this and taking into account all the skyscrapers, are you predicting that we may fall into either a very another recession or even a depression? Yes, you are. Oh, I am. Gary, the stock market has been engineered by the central bank. It's clear. I mean, the interest rates have been artificially low near zero for seven years. And much of that has been also several years of quantitative easing. And those interest rates, if you ask your financial advisor, those things translate immediately into higher asset prices. So stock prices are artificially high, bond prices are artificially high, and real estate prices are artificially high. And, you know, I just don't know how long the central bank, the Federal Reserve can maintain zero interest rates in an economy. This is very unusual. This is, I mean, it's time to wake up and smell the roses basically here. This is a very unusual. It's never been done in the history of man where central banks around the world are suppressing interest rates. And now fall, since we last talked, governments are able to sell government bonds in many countries around the world at negative interest rates. Really? So people, yeah, people are in Germany and in Switzerland are paying that money in exchange for bonds that are going to result in a negative return. And primarily, this is people with lots of money who don't want to hold the money in banks because of the legislation that's been passed and the rules that have been made that where bank deposits and the bank deposits in major banks can be converted into shares in the banks. If the bank gets into a crisis situation, the rules now state that basically those large depositors can have their deposits in currency converted into shares in the banks. And so people don't want that. And so they're willing to buy government bonds with negative interest rates on them, which is unheard of. I mean, it's never heard of it. It's in a normal economy that an impossibility, but in this rigged economy, apparently anything goes. Wow. Well, Mark, I mean, I hate to leave it on such a sour note, but I'm very, very concerned. And I guess we just hope and pray that somehow, at some point, they start to uncap or release interest rates, slowly bring them up or whatever. I don't know if they'd want to do it, all fell swoop one time. And then slowly allow them to rise. Do you think that we might be able to do that? Do you think there's a possibility they'll do that? I think, you know, if they set a major economic reform package into motion, which tells people that interest rates are going to be normalized, that government budgets are going to be balanced, and that the impact of government on the economy is going to be reduced. Yes, I think that that would be very settling and stabilizing. And these are some of the things we're going to be talking about next week at the Austrian Economic Research Conference here in Auburn, Thursday, Friday, and Saturday. And your audience can tune into the major lectures on Mises.org. They'll be live for the major lectures, not for the breakout sessions, but for the major lectures. And tune in live and see what Austrian economists are from around the world are what we're up to nowadays. Although I think actually the side conversations are going to be the most interesting. I bet they are. But let's repeat that. Go to Mises.org, Thursday, Friday, and Saturday of next week and tune in and find out what these Austrian economic leaders are talking about at the Ludwig von Mises Institute, right? Absolutely, Bob. Terrific. Well, again, as always, Mark, it's always so interesting to talk with you because you bring a perspective that a lot of people don't even think about bringing. So very interesting conversation. And I thank you as always, and we'll talk soon. Thank you, Bob. I enjoyed it.


Welcome back to the lectures on the Principles of Quantum Mechanics. The series of lectures is being offered to you through the national program on Technology Enhanced Learning NPTEL and my name is Mangala Sundar Krishnan. I am a professor of chemistry in the department of chemistry Indian Institute of Technology, METROS. In the series we have been looking at in the first part of the series we have been we have been looking at some of the elementary mathematical aspects related to the Schrodinger equation and also basic quantum mechanics and in that context in the last lecture we looked at first order differential equations.  Therefore, I shall state the problems for those of you who want to do the solutions by themselves and not listen to the lecture. You can always refer to the lecture for details when you need to, but you can skip this lecture and go to the next one for some details on the second order differential equations and the power series method for solving mathematical problems in quantum mechanics. So, this is more like a hand holding or a problem solving lecture and I shall state a few problems that we want to consider in this hour. We will split this into two 20, 25 minute sessions. So, that you can have a break at some convenient point that I have decided that that is a convenient point ok. The first set of problems very elementary to slightly more involved. Let me write a simple differential equation d y by d x minus 10 y is equal to 0 and another problem d y by d x minus say phi y is equal to 6. These are given as problems for the background quiz on the website. So, they are not with any different from what is already there and the third that we will look at in this group is d y by d t minus y by t is equal to 0 ok. So, there are 3 yeah these 3 problems first let us look at that. Obviously, these are trivial I think you must know that this is a simple exponential solution. So, you can write that d y by y is equal to minus 10 d x which gives you immediately y of x is equal to a e to the d y by d x is plus 10 x. So, it gives you a e to the 10 x where a is the arbitrary constant. So, one condition will tell us what the value of a is that condition can be on the y or on the y prime or whatever it is at a particular value of x. So, there is one arbitrary constant since it is a first order differential equation. The second problem d y by d t d x d y by second problem d y by d x minus phi y is equal to 6 is amenable by a simple method of finding this as a single function which you can immediately see that it is d by d x of e to the minus phi x y of x. The differential of this will give you this part the on the left hand side and since you have multiplied the whole equation by e to the minus phi x in getting this form this is 6 e to the minus phi x ok. All that I said was multiply by e to the minus phi x and that gives you this ok. So, the solution would be y of x is equal to e to the plus phi x integral e to the minus phi x d x 6 times and the answer that you would get is y of x is equal to minus 6 by 5 plus a constant times e to the phi x ok which if you substitute in the differential equation you will get the equation back by phi. The third problem d y by d t minus y by t is equal to 0 is a separable problem because it is d y by y is equal to d t by t and then integration gives you a constant. So, you will get y is equal to a t it is essentially l and y is equal to l and t plus c. So, it is a constant it is a straight line in t ok. So, this is the first set of problems. Let us look at a slightly more involved set of problems involving the other methods. Let us look at the other set being d y by d x is equal to x square plus y square by x y solve for y. Second problem is d y by d x is equal to y by x plus square root of x y and one more d y by d x is equal to 2 plus y e to the x y y 2 y minus x e to the x y. So, in all these three cases solve for. So, let us take the first one this you recognize is a homogeneous of order 2 of degree 2 the equation d y by d x is equal to x square plus y square by x y. Therefore, the standard method that we learnt in the last lecture is that let y by x be u and y is equal to u of x d y by d x is given by u plus x d u by d x.  So, you get 1 by u therefore, the solution is x d u by d x is equal to 1 by u or u d u is equal to d x by x integrate to get u square by 2 is equal to l n x plus c and u itself is y square by x square therefore, you have the final answer y of x is equal to square root of 2 x square l n x plus c prime c x square that should be a 2 perhaps and u square by 2. So, that should be a 2 here also. If you have an initial condition or if you have a condition for y given the value of x you can obtain the value of the constant c is a constant. The second this is the first of the three problems yes. So, the second one d y by d x is equal to y by x plus square root of x y this also has the same form it is homogeneous and the degree is 1.   Therefore, the differential equation is minus d u by u root u minus d u by u is equal to d x by x and the answer turns out to be you can solve this immediately. This is square root of u the answer is 2 by root u for this and the answer here is minus l n u this is equal to l n x plus c giving you the answer that 2 root x by y is equal to l n y plus c c a constant. The third problem because the way it looks like d y by d x with it appears non-linear y e to the x y 2 y minus x e to the x y. So, we try this by writing it as m d x plus n d y is equal to 0 and the c if dou m by dou y is equal to dou n by dou x in which case we have a function of x and y which is a constant as a solution ok. So, if you rewrite this differential equation you have 2 y minus x e to the x y d y plus minus 2 minus y e to the x y d x is equal to 0. So, this is assuming this to be n and this to be m let us look at the derivatives is d y. So, I have to look at dou n by dou x and that is the partial derivative with respect to x here which will give me minus e to the x y minus x y e to the x y since differentiating this only with respect to x only ok. And dou m by dou y you look at the derivative here is minus e to the x y minus the derivative of the exponential with respect to y is x y e to the x y and therefore, we have this equation dou n by dou x is equal to dou m by dou y and therefore, there is a function f of x of y which is a constant such that d f is dou f by dou x given y d x plus dou f by dou y given x constant d y and that is 0 and this is your m this is your n and all that we did was dou squared f by dou y dou x is equal to dou squared f by dou x dou y for well behaved functions ok that is what we used. Therefore, the immediately we know that integral m d x with respect to x will give you the integral of minus 2 minus y e to the x y d x only and the answer will be not a constant, but any function of any arbitrary function of y because we are only integrating with respect to x. So, the answer we can write as minus 2 x minus e to the x y plus a function of y plus other constants, but that we do not want to include right now and how do we determine this we have to find out what n d y gives which will give you a function of y and an arbitrary function of x you have to match these two in order to get the final answer. So, the n d y integral is integral 2 y minus x e to the x y d y which is given as y squared minus e to the x y plus g of x. Now, you can see that here is one solution and here is the other solution and the function f of x y is the correct combination of these two such that f of x y satisfies the above equation. So, you can see that there is a y squared here and which is the h of y and e to the minus x y is common to both of them there is a g of x which is minus 2 x and therefore, the answer is f of x y is y squared minus e to the x y e raise to x y minus 2 x and of course, any arbitrary constant to that because you can always add a constant here you can always add a constant here and then well make it c double prime some arbitrary constant. So, the solution should be y squared minus e x raise to x y minus 2 x plus a constant. So, this is one way of solving the differential equation. So, let me complete this exercise with one more set of problems namelyverifying the integration integrating factor or these exact equations. Question given is y d x minus x d y and the factor minus 1 by x square is it an integrating factor. You can see that if this is m d x and n d y you can see that dou m by dou y is 1 dou n by dou x is minus 1 and therefore, you see that this is not equal to each other I mean they are not equal to one one other and therefore, you see that this is not an exact equation. However, if we multiply this equation y the if we multiply this quantity by minus 1 by x square you have minus y by x square d x minus plus 1 by x d y then you can see that this is a perfect differential. Therefore, if you have an equation like this is equal to 0 then what you do is you multiply this equation by minus 1 by x square and then make this into a perfect differential and solve this. This is easy to verify because if this is m and this is n dou m by dou x is dou y I am sorry dou y is minus 1 by x square and dou n by dou x is again minus 1 by x square. So, that is an that is the integrating factor. Likewise,there are two other problems is 1 by x y an integrating factor for the quantity y d x plus x d y actually y d x plus x d y itself is a perfect differential it is d of x y therefore, dividing it by x y obviously, will make it an integrating factor anyway. So, again make it perfect differential. So, we do not need it, but it is definitely an integrating factor for this problem because what you have is 1 by x d x plus 1 by y d y and you see that the derivatives in both cases will be 0 which is obvious because when you divide x y by x y you have a constant therefore, the derivative of a constant is 0. So, if f of x y is x y then you can see immediately d f is given by this quantity right. However, if you have a minus sign as you have here you do need an integrating factor. So, these are some simple examples of thefirst ordered differential equations. There are many many suchexamples and I probably will givean assignment on the websiteplus some handwritten solutions of those. So, those of you who want toget familiar with a little bit more of these operational methods of elementary first order differential equations please refer to the NPTEL website under my course the Principles of Quantum Chemistry or Quantum Mechanics ok. You can search for myname and you will see the courses that are there on the website. So, let me stop this part of today's lecture here have a break and then what we would do is to start with elementary second order differential equations and then continue that with somesimple examples in the next lecture. So, that the three lectures the onethe the preceding one this one and the next one hour lecture one and a half hours whatever together will form an elementary introduction to differential equations solution for those who want to study quantum chemistry and whose mathematical backgrounds are not I mean formally well defined in the areas of mathematics. However, if you want to learn these things more thoroughly it is important for you to go back to the books and read some of the proofs the theorems and the existence uniqueness of these solutions etcetera. A lot of these is also done in the NPTEL program under the mathematics and under the physics department. This is a hands on help to the chemistry graduate and undergraduate students who would come across quantum mechanics for the first time and are often disturbed by the fact that they have to solvesomewhat difficult differential equations and then never use them in practice later on because almost all the learningmethods and the practice is computational and numeric. However, it is important to appreciate some of the elementary solutions and particularly exact solutions of quantum mechanical problems. So, let me stop here we will continue this after a break.Welcome back we shall continue this with the second order differential equations. Again we will only do what is minimally needed to take us through the solutions ofthe quantum mechanical problems of harmonic oscillator and the hydrogen atom. So, most of you are familiar with these elementary differential equations second order d square by e x square y plus some constant k squared y of x is equal to 0 or d square y by d x square minus k squared y is equal to 0. Well, the sign difference is notthat simple, but it leads to basically different solutions I mean different the solutions are of completely different nature. This gives you oscillatory solution and this gives decaying, growing solutions. So, exponentials both, but very different nature ok. Then the other types of second order differential equations the general linear second order differential equation d square y by d x square plus a function p of x d y by d x plus another function q of x y is equal to 0. This is homogeneous linear second order d e and the allied inhomogeneous equation d square y by d x square plus some function p of x d y by d x plus q of x y is equal to g of x inhomogeneous or non-homogeneous this right this is non-homogeneous equations. When p and q are constants in p q or p q or constants then there are simple methods for solving them. When p and q are not constants, but functions and also sometimes a function preceding the second derivative itself. For example, well l of x d square y by d x square plus p of x d y by d x plus q of x y is equal to some other function g of x. We have far more general and much more detailed methods of solving them. One way is to look at this l x by not looking at it, but by bringing it as a divisor, but then there could be problems if l of x goes to 0 at some particular value of x which is in the domain of the solution of the equation. And those things are called the singular points and solutions of differential equations with singular points is one of the most beautiful and most thoroughly studied branches of mathematics. And a lot of that is also useful in quantum physics and in of course the entire branch of physics and engineering. We will not do any of those things at the moment. I mean you can take them up later, but probably we will do a little bit of these. And then proceed directly to what is known as an infinite series method or power series method for solving the differential equations, particularly second order differential equations. First simple solutions d square y by d x square plus k square y is equal to 0. It should be immediately obvious that y of x has the solution a cos k x plus b sin k x twice differentiating this you will get this you will get that equation ok. Another way of writing the solution is instead of a cos k x to write to this as e to the i k x plus e to the minus i k x by 2 plus b by 2 i e to the i k x minus e to the minus i k x which gives you two new constants c exponential i k x plus d exponential minus i k x. So, this is the complex representation for the solutions to the differential except that that c and d now can be complex constants or real constants you can see immediately b by 2 i plus a by 2 or minus whatever is defined by c and d. So, it is possible to write this. This is an important equation that we will see in a particle in a box and particle in a 2 d box and so on. So, we will have to keep this in mind ok. We will see this again. Another basic differential equation is d square y by d x square minus k squared y is equal to 0 and this has clearly only one solution y of x is equal to a e to the k x some arbitrary constant b e to the minus k x ok.  So, you will get exactly this equation, but this is exponential growth and exponential decay this is not oscillatory as you have in the previous case ok as you have in this case ok. So, the sign is quite important and the fact is that k square is real if k square is imaginary. Obviously, this will become negative and so you have solutions if k is imaginary. Obviously, this is e to the i some real number and the solution goes back to this elementary. Now, the type of equations that we will study for in this course or the following equations ok. One Legendre differential equation of order alpha and this is a special equation solved by obviously, the legendary Legendre one minus x square d square y by d x square minus 2 x d y by d x plus alpha into alpha plus 1 y is equal to 0. In this course I will not solve this equation, but it is important in physics and in mathematics it is known as Bessel's equation of order n given by x square d square y by d x square plus x d y by d x plus d y by d x plus d y by d x plus x square minus n square y is equal to 0. Please note that it is multiplied by the second derivative is multiplied by x square. Therefore, x is equal to 0 is a singularity and one has to be careful about writing the solutions of this or solving this equation ok. We will not do this in this course, but it is important to remember that it is an extremely important equation in physics and mathematics. In this course we will see this equation log air differential equation for the solution of the radial part of hydrogen atom electron problem. The log gas equation is given by x square d square y by d x square plus 1 minus x d y by d x plus alpha y is equal to 0 and associated log air differential equation x square d square y by d x square plus 1 plus k minus x where k is non 0 d y by d x plus alpha y is equal to 0 and lastly oh there are two more equations. We have might differential equation d square y by d x square minus 2 x d y by d x plus 2 alpha y is equal to 0 ok. So, this we will solve explicitly in this course, this we will solve explicitly and also the Legendre there is also an associated Legendre equation associated given by 1 minus x square d square y by d x square minus 2 x d y by d x plus alpha into alpha minus alpha plus 1 minus m square y is equal to 0. So, we shall solve these types of equations in detail later on, but let us look at simple theory of second order differential equation. Well, simple examples I would not call it theory, simple examples for second order differential equation. If we consider homogeneous equations with constant coefficients these are easy to solve meaning the equation a d square y by d x square plus b d y by d x plus c is equal to 0 homogeneous constant coefficients a b c then the and a is not equal to 0. So, if you have merely d square y by d x square a is equal to 1 otherwise any number that goes with it then it is easy to solve this equation by having this operator defined d as d by d x and the operator d square as d square by d x squared and you can see that a function which does not change through differentiation to any order is an exponential function. Therefore, we might propose as a simple solution to this differential equation an exponential function and then if we substitute that exponential function with some unknown constants in there we can probably transform this equation into a linear equation not a linear differential equation in terms of those coefficients and then determine this coefficients and since you have second derivative here and first derivative here when you take the derivative of exponentials with coefficients the most you will get or the powers of the second powers of the coefficient and the first power of the coefficient and so on. So, this will turn out to be a simple quadratic equation in terms of those coefficients and quadratic equations have analytic solutions in terms of the roots of the quadratic equations. So, it is very easy and very quick to understand the solutions as to why they are important. So, if I have an equation like a y double prime as I have written down b y prime plus c c y is equal to 0 and if I propose a solution y is equal to a constant times an exponential say k x we could choose minus k, but it does not matter because when you solve for the y using this form you will get different values for k. So, if you do that d y by d x is k y and d square y by d x square is k squared y and so you will immediately get the equation a k squared plus b k plus c times y is equal to 0 and since y of x is not 0 everywhere arbitrarily it is non 0. You have the quantity a k square plus b k plus c is equal to 0 and this gives you the solution for k as minus b plus or minus square root of b squared minus 4 k c by 2 a. So, you get two values for k k which you can call it as k 1 and k 2, k 1 is obviously one of the roots minus b plus square root of b square minus 4 a c by 2 a and k 2 is minus b minus square root of b square minus 4 a c by 2 a and given these two roots the general solution for the differential equation will be a e to the is equal to a e to the let me just get this right e to the k 1 x plus b e to the k 2 x where k 1 and k 2 are given by this. So, that becomes the solution of the differential equation. So, when you have constant coefficients for a second order differential equation it is nothing other than a simple quadratic form, but interesting thing is when k 1 and k 2 are equal or when k 1 and k 2 are not real, but complex then you have to worry about some special forms. If k 1 and k 2 are not equal to each other real or complex the solution is easy to write down. So, the best would be to illustrate this using a simple example. Let me take an example for solving such an equation. The examples are there in the assignments in the website, but let me see if I have one here. Yeah. So, let us take two examples and we will illustrate the differences between them. For example, one problem that we have is y double prime minus 4 y prime minus 5 y is equal to 0 ok. You have just seen that this with the substitution y is equal to a e to the k x this essentially goes down to k squared minus 4 k minus 5 is equal to 0 which is easily factored into k minus 5 times k plus 1 is equal to 0. Therefore, k is equal to minus 5 k is equal to 5 or minus 1 mean that is a quadratic equation solution anyway. So, you have immediately the solution of the differential equation is y of x is a e to the 5 x plus b e to the minus x easy real different values for k. Now, an interesting variation of this would be to solve this differential equation y double prime minus 4 y prime instead of minus 5 let us put plus 5 y is equal to 0. Then if you factorize this what you would get is k squared minus 4 k plus 5 is equal to 0 and this the roots have to be solved through this discriminant form. So, you will get k is equal to 2 plus or minus 2 i b minus b plus or minus b square root of b square minus 4 a c all those things you get 2 roots.  But you have an oscillatory term with the growing exponential that is because the leading term or the real part is positive. If the real part is negative you will have a decaying exponential but oscillatory. So, this is what happens if the coefficients k 1 and k 2 are imaginary or complex. What happens if they are both the same? For example, y double prime minus 4 y prime plus 4 y is equal to 0 which when you factor out it is k square minus 4 k plus 4 is equal to 0 or it is k minus 2 square. Therefore, k is equal to 2 is the only solution is the only answer. Does that mean that y of x is equal to a e to the 2 i of x minus 2 x is the only solution? No. Does that mean? No. Because it is a second order differential equation we expect two general solutions or general solutions with two arbitrary constants and therefore, there is another form that we need to worry about to in instead of explaining why that form is let me give you that particular form namely we write this y of x as a e to the 2 x plus a general form b times x e to the 2 x that it will turn out to be only x and not x square or x cube etcetera you can easily see that. But if we have x to the e to the 2 x also will be a solution to this equation. So, let me just do this in a slightly different way y it is x that is a sorry write this y double prime minus 4 y prime plus 4 is equal to 0.  Where f 1 is a solution is a solution ok. So, we have already found out f 1 to be a e to the 2 x. So, if there is another solution f 2 what we need to do is to try out f 2 is equal to g of x e raise to 2 x we will keep that because exponential is always there for all the second order constant coefficients differential equations. But we need to now determine the arbitrary function g of x and in a minute in a few minutes you will find out that this is nothing other than x ok.  Now, the derivative if you substitute for d f 2 using the g of x e raise to 2 x what you will have is d by d x minus 2 d g by d x minus 2 d g e to the 2 x is equal to 0 ok or d square g by d x square d g by d x of e to the 2 x minus 2 d g by d x of e to the 2 x is equal to 0. And therefore, what you have is d g by d square g by d x square times e to the 2 x is equal to 0 because this 2 will cancel with this 2. I have done only differentiate this whole thing using d by d x this one. So, what you have is d square g by d x square is a is equal to 0 which means d g by d x is a constant or g is a constant times x. So, you see that it comes out immediately when you substitute a general form a general form f of x this is the general form that we used sorry the general form that we used was f 2 of x g f x. If you substitute that in the differential equation you immediately get this answer ok. Therefore, the general solution for the differential equation is one is the a e raise to 2 x the other is the x e raise to 2 x. And this is the form for the homogeneous differential equation with constant coefficients such that you get a perfect square for the differential forms like this. So, there are many tricks of the trade here and the job of a theoretician is of course to keep as one of my post doctoral colleagues senior colleagues when I was under training when I was associated with him kept on telling that you have to collect the tools and your toolbox should keep on piling up mathematical techniques for theoreticians. One should continue to to learn and keep them ready and use the necessary ones. So, differential equations are are many methods the homogeneous linear second order differential equation has some very simple solutions as we have seen. In the next lecture I will take this a little bit further and talk about the general solutions for the differential equations with the coefficients which are not constants, but functions like the p and the q that we wrote down. And then probably given some time we will also look at one important method for chemists namely an infinite series representation of the solution and then solving the differential equations using an infinite series. We will assume the infinite series to be convergent all those details will not be discussed. We will come back to the differential equations in the next lecture. Until then thank you very much.