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Consider the following initial value problem,')y" + llzy' 47y = 0, y(0) = 4 y (0) = 0.Note: For each part below you must give your answers in terms o...

Question

Consider the following initial value problem,')y" + llzy' 47y = 0, y(0) = 4 y (0) = 0.Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimalsa) This differential equation has singular points atNote: You must use semicolon here to separate your answers_Since there is no singular Point atI = 0, you can find norma power series solution for y(z) about =0, i.8.,y(z) am 1m m=0As part of the solution process you must determine the recurre

Consider the following initial value problem, ')y" + llzy' 47y = 0, y(0) = 4 y (0) = 0. Note: For each part below you must give your answers in terms of fractions (as appropriate), not decimals a) This differential equation has singular points at Note: You must use semicolon here to separate your answers_ Since there is no singular Point atI = 0, you can find norma power series solution for y(z) about =0, i.8., y(z) am 1m m=0 As part of the solution process you must determine the recurrence relation for the coefficients Gm: Enter your expression for Gm+2: am+2 (m"2-12*m+47)(m^2+3*m+2) am



Answers

m3.1 obtain the solution of the following differential equations

In this question we have to find a linear homogeneous differential equations with constant coffee. Since whose rules haven't given us M equals 000 minus two minus 23 out and minus three. Health will first consider the first three zeros that have given us routes since Amy called zero. You can see the M plus zero zero again and plus zero zero again and plus zero. So you can see that Michelangelo and will multiply all these three. We'll get em kiwi called zero. Now let's just see the second one that is equal to -2. And again in cold minus. So if M equals minus that, we can see that M plus triple zero. And again, because the rules are minus and minus. So we multiply these two. That is plus 1 10 plus two. We'll get down to +00 plus two will be Now for the hard part that is m equal to three out and again, simple to minus three out. So for chemical three out and Ministry out of Visual and for me called minus helpless three out of So we'll multiple at this represents that is -3 out into the first three out and we'll get 0-00. So we'll get when you multiply these two will get them square -9. I spell it and we know that I ought to swear you called-. So we'll get em effortless 90 collagen. Now again will multiple all these 3% that is this one this one and this one and we'll get zero that is thank you And two members to elsewhere. And um start last night. Now we'll just open the brackets plus two. Open this square square one. That is m plus two holes. Whether it will get M squared plus four plus four. Now again, we'll open this bracket, keeping them to we'll get em fourth plus nine M squared plus four M. Two. Statistics name plus 26.0. Now again. Well, open up the record for the final time. That this will get empty about seven plus four and six. They're starting them to fight the strategic 4 36 M two Q. Is equal to zero. No, just try to remember this. We know that when if this is the given different allocation, we try to replace by double death by M square and wider by him. And why simply were one to get out of the equation and then solve for the roots and get the differential equations. Also the different different person. Similarly, we'll go and solve this one, but we'll go reverse this time. But if it's uh auxiliary question is this one, what will we tell modernist different person. This will put And to those seven years. What? You were seven and into a success right? Or 6 25 25 24 hours prior to the fourth and M two or three years? And this will be there. Our homelessness differential equations with Western conference whose rules will be this that is mhm 000 -2 -2, 3 out of zero, and this will be the auxiliary person. Yeah. Mhm.

Who we've been given this differential equation right here. So I'm giving you uh step by step solution to this. Uh All the points are very clear, so I hope you understand this, wow. So of course we're just gonna do differential of that, which is this and then the differential of uh the same thing. That's a double to friendship. We're gonna get that and then we're gonna substitute back into the equation one. Okay, This is the first differential second differential within the next slide. This is the substitution for that. Okay? Uh So here you realize that uh this X and minus one and minus one. Excuse me? You know, my eraser? So and minus one and the same as X and over X. Okay, so this X right here is gonna cancel out this X. Okay, we're gonna get this. So by implication the series the submission from and it was 02 infinity is gonna give you that. Okay. From the previous slide. So once you have that uh we cannot use the recurrence relations where to a two equals zero. So that gives a two equals zero. So from here we're gonna be testing for and of course, 123 and four. Okay with this relation. And uh that is how we're gonna draw our conclusion. So from there you realize that why it was that. And here you can do substitution from the previous slide. Okay? And then uh you're finally going to get that. Okay? Thank you very much.

Okay, solve this differential equation Using the power series. So I'm gonna let my power series B in equal 02 infinity A sub N X to the end. So then it's derivative is in equals one to infinity A psa ban in X to the n minus one. And then it's next derivative. Don't forget to be raising the in here from 0 to 1- two. A seven in -1 X to the N -2. All right now write the equation. So the equation is I mean plug that into the equation and equals to a sub n in and -1 X. to the N -2 -2 x times the original function. So that will be an equal 02 infinity A sub N X two. N plus one equals zero. Mhm. Okay, so that's what we're going to try and solve. Okay, so the next thing you do is you make sure this exponent here and this exponent here are the same so that we can combine these two terms together. Okay, I don't know how you learn to do it. This is how I do it. Okay, so I want this in -2 to be K or M. or whatever. So I'm just gonna say that let em equal in -2 Then that means in plus two is in Okay, so this turns into M plus two equals two to infinity A. M. Plus two. M plus two M plus one next to the M. Plus Yeah. Okay, I need this one to be the same thing but just so I don't get confused. I'm going to call it cave for now and then I'll switch it Like a equal in plus one than K -1 equals in. So this turns into K -1 equals 0 to infinity eight K -1 X. To the k. Okay, so in plus two equals two. That's the same thing as M equals zero. So I'm just going to change that one and k minus one equals zero. That's the same thing as K equals one. So this one over here on the left starts at zero and this one starts at one. So that means I need to get the zero term off it here and out of the way so that I could add them together. Okay? So if I plug zero in I get A sub two times 2 times one Times X. to the zero plus the rest of this series, starting at one. Now because I took the 01 off there, I'm gonna switch this to him now because it's just a dummy variable anyway. Okay, so here's what we know now we're gonna start with a zero equals some a zero. We don't know it's just something And a one equal some a one. Okay and then this has to equal zero because there's no constant term over here on the right hand side. So two times one times a two has to equal zero. That means a two is equal to zero. Okay now what we have to do is figure out what the rest of our that's what the rest of this is for. So I'm gonna put it into one series. It's a. m. plus two and plus two M. Plus one. Oh I've I accidentally changed what I accidentally change this minus here to a plus. So that should be minus and that should be minus. Let me get the eraser room. Okay that should be a minus there. Didn't magically change minus minus minus AM -1 X. to the M. equals zero. Okay now all of these things have to equal zero because there's no X terms no X square it's no excuse or whatever. So A. M plus two M. Plus two N. Plus one Equals AM -1 or a. M. Plus two equals a M minus one over M. Plus two. N. Plus one. Okay so now now we can find all the rest of them. Okay I'm kind of a baby here. I'm going to find a three. So that means I'm doing M equals one. So it's a 0/3 times two. And then 1/4 that's where m equals two. That's a 1/4 times three and then a five that's where M equals three. A five. That's a 2/5 times four. But that 10 because remember over here A two is zero. Okay I'll see you two more. A six that's where M. Is four which a three A six M. s. for a 365. But a three is a 0/3. two. Okay, so this is a zero over 6532. That's so weird. A seven has where M. is five. So that's a 4/7 times six. So that's a one seven times 6 times four times 3. Then the next one It's going to be zero. Okay. So now here's what we got we got two series here. Half of them had three Really? Half of them have a zero in it. Uh huh. Half of the non 01 to have a zero in it. The other half of the non zero ones have a one in it. And the other series is just all zeros. So the a zero series goes a zero one plus A three is the first one, 1/3 times two X. To the third and then plus 1/6 times five times three times two X. To the sixth plus. All right, let's guess the next one. Uh Clearly the powers are Multiples of three. So X to the nine they always start with the same multiplier. So nine times 8 skip one 6532 plus dot dot dot Plus the ones that start with a one and the a one term member goes with eggs. Don't forget that. And then the next one is this one so plus 1/4 times three X. To the fourth Plus the next one is X to the 7th. 1/7 times six times four times three X to the seventh plus dot dot dot. We could guess the next one. That's the answer. Except for we have some initial conditions why zero equals one. So if you plug that in, everything goes away except for a zero. So that tells you a zero is 1. Okay. Also we know why prime of zero is one. So if we take the derivative of all this um that will go away. That will go away when we plug zero in all those will go away. Oh except for this one right here when you take the derivative that will be a one. But everything else will have an accident. So then we also know a one is one. So then the answer to the question is why equals one plus x cubed over three times 2. Oh it was ready this way. One plus X. No X squared term X cubed over three times 2 X to the 4th over four times 3 plus no X to the fifth term X to the 6/6 +532. It's always disappointing when you get here and it's not a series that you recognize. But I mean it's not our fault. Whoever wrote this problem didn't give us a good one that we could know. Okay if you want to check it, take his derivative, take his next derivative multiple. Do what you're supposed to do. I mean, you know, plug plug it into the equation and see if it works. All right. I hope that helped. I try to be super neat and sorry about the little mine I signed mistake.

In this question we have to find a solution for the differential equations given to us. There are four questions. The first one is they were well the toxic was too X plus one divided by where do the power four plus one. And we have to find it for Mexico's leader and why close to zero. So first of all I am going to separate the where he was here. We across multiple. I then why with about four plus one into the Y equals two. X plus one into the X. Okay I'm going to integrate both the sides. Then get rightly about five by five. That's why it was to access whereby to less X. Let's see. Okay this is the integration here and I read it a constant. Because I have done here indefinite integration now I have to use this condition execute zero. Why it was zero who calculate the value of constant. So I protect the close to zero and why it was zero here. So I put this and then I get zero plus zero equals to zero plus zero. Let's see. So from here you get the value of C. So we used this where you will see and after that I get the final solution that is y to the power five by five. Yes. Why it was to access square by to bless exp Let's see. Okay. This is answer for the first question. Now we have to find a differential regulation for another question solution for the differential equation? For the second question. The second question here is actually why minus whose idea it was to access? We're leaving. Okay. So first of all we are going to divide by dx. So we divide by two years. So it attacks divisive ideas minus y. It was to access square and you're in this part, you can see that you have here. Do you buy Y. Dx. Ok. If I divide one dx. Now I can say that I I'm going to take this divide by the ex common and it's taking it on the same side. We can say that derive ideas equals two X minus access where it was too white. Okay now we have to yeah separate the variable. So you get your divided by like crystal the X place x minus x squared. Now after that I can say that I have to integrate with the sides. So your integration up divided by Y. And here I take minus common and the except one affected by vaccine. Two x minus one. Now we can say that this can be better next day. Bye bye bye. It was two minus. Or if we make here factors then we have to survive here. Okay one divided by x minus one minus one. Divided by excess. These are the factors that are formed here. Okay you can see clearly now after all we have to integrate all the yeah. Thanks. So be by my wife's integration is the island of what equals two minus. Its integration is Eleanor of x minus one and minus Eleanor X. And let's see so give students see we have we're dealing oxy because I can say that since all the terms are in Ellen so I'm adding LNC no area only than 13 11. So I take this one the selling only dioxide website. Forget Elena boy La salina affects minus one minus Eleanor. Thanks. Okay it was to L. N. C. And I have like the poverty of natural law. So it provides me Ln of Y into x minus one divided by X. It was to L. N. C. Okay now after that we are going to cancel out. And then you get why do why into X minus one? Divided by its It was to see and this can be due to illness lying to x minus one. It was to see X. Okay, this is the solution for your second question and after that we have the third question. Third question says that we are going to great extra do a minus why D x minus of the root of X squared minus y square into the X equals to zero. So first of all, I converted it into the standard form means I am going to divide their dx with all the terms and it can Britain us. They were years. It was too, I can say that a vote of one minus Y by X. Police work plus Y by X. Okay, after converting it into the standard form now I'm going to substitute here. Why Buy X? It was two weeks this so that's why it was two years and yes there during the day. Bye bye. The X equals to replace ex BBB ideas. Okay so we made these substitutions in the about differential equations so this provides maybe plus excellent baby by the exit was to go to one minus we swear let's all week. Okay this answers this week and now I'm going to separate once again the variables here. So after separation they have one root of when minus ways where it was to be expo next. Now we have to integrate both these years. So integration of one of one road one minus the square is sign anniversary equals two L. N. O. X. And after that let's sell enough C. Okay now I put the value we again value of these. Why buy extra signing words Huaibei X. It was to Eleanor C. X. This is the answer of the the hard part and there is remaining another part. Also fourth question. Okay so the fourth question says that we have to integrate the Y. By the exit to us too. Access square last Y squared divided by next one. So I have to divide it by access. Where if we divided by X. Is where then it changes into I can say that one plus why by ex police were divided. Well why by X. Okay now we substitute here again. Why? By exit was two weeks. This shows that why it was to re X. And its derivative is divide by the execution with classics. Divisive ideas. We replace this in our differential equation so that we get U plus X divided by D. X equals two. One plus the square divided by B. Okay my way. Right it has replace X. Dy dx sequester one baby. Let's go be okay this cancels the RV and now we have to make variable separation here so we write it as well. D. V. Equals to the Xbox. Now we integrate both the sides the integration of these risk whereby to and integration of one of the one access island affects and bless Helenowski. Okay now after that I have to put again the value agree. That is why by yes. So it changes into one by two. Bye bye X. Holy spirit Western Eleanor six. If we remove over Ireland then it provides you C. X equals two people. The power wife whereby two X six word. Okay so this is the final solution of the fourth question and there is no need of the removal of Ellen. You can also take the humble one is answer also what are the answers of the second question is your last question of it. Thank you.


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