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(60 points) Use power series t0 solve the initial-value problem 4ry' 8y = 0y(0) = 0y (0) = 1Answer: yr2nrn+...

Question

(60 points) Use power series t0 solve the initial-value problem 4ry' 8y = 0y(0) = 0y (0) = 1Answer: yr2nrn+

(60 points) Use power series t0 solve the initial-value problem 4ry' 8y = 0 y(0) = 0 y (0) = 1 Answer: y r2n rn+



Answers

Use power series to solve the differential equation.

$ y'' + x^2y = 0 $,

$ y(0) = 1 $,


$ y'(0) = 0 $

Were asked to solve this differential equation why prime minus one equals zero. It was in power series but first of all we can see that the solution is just an exponential function. Um And so we should uh you know we can get that but well to use the power series, we should basically wind up with the power series. That is a representation of the exponential function. So um I use this summation notation here because really only care about this. Lower this lower bound of the summation that the upper bound always goes to infinity and we always are something over end. So I just use the simple or notation to clean things up. So this is the summation of N goes from zero to infinity. I've seated the end actually end. So we assume that for why kick the derivative we get summation from n equals one to infinity. You see to the end and X to the n minus one tonight. So what we have here um and we could change this to zero but it doesn't really make a difference because you know, if any was zero then this would be zero and it wouldn't add anything to the summation. Take these two things, I think these two expressions and plug them into here and we get this. Now what we want to do is we want to get everything, we want to get all these exits to the same power, so we can factor them out so we can shift end, so we can shift in by and at one to end. So we get basically let end go to end plus one. So we get seated the end plus one and plus one next to the end. And now we're going from zero 02 infinity and noticed that, you know that that is the same. These two summations, are they safe? And then we pulled the summation out because now we have the same summation from zero to infinity. And now you can see we have X P N. And every time. So that will factor out assuming X is not equal to zero. Um because this needs to be true for all access. So basically if this is true for all access, then these coefficients have to be um the the coefficient has to be zero. So that means end plus one times seat at the end. He has the equal seat at the end. Let's give us a recursive equation for seat at the end. So see to the end equals one over n plus one times see to the end plus one equals one over n plus one seat at the end. Yeah, So C one when an 80 we have C one equals Xena. When I think was one we have C two equals C 1/2. Which is he not over to when And of course to we have C. Three or C 2/3 which is C. Two or C not over three times two times one. And we can see the pattern emerged that see to the end it's just saying that over and factorial. So we take that and plug it back into here and we get Y equal sina times the summation and goes from zero to infinity of X to the end. All over N. Factorial. And this is actually the this is the power series for either the X. So if you would take the power to the taylor series of either the X. Um you would get, you know this expression here. So again, it's only good for around X equals zero and it will start to diverge. S. X gets too large if you don't take a sufficient number of terms here.

Today we're going to solve program number nine. Yeah, my dad realized, minus exploitation Mine, That's why equals zero we used by equals. Take mark Any go to zero to infinity. See, an existing wild eyes equals like what they need goto one to infinity de en in doing existing minus one by double that she was sigma unequal toe do to infinity See and into in into n minus one express doing minus two by a double digit was sigma and equal to zero to infinity See and plus two into n plus two into n plus one existed in expired ash It was we got like sigma and a good 1 2020 CNN doing existed So we can write like sigma in equal to 0 10 20 see and plus two into and plus two in the end plus one into express two and minus sigma unequal to want to infinity CNN do and Hindu express two and minus sigma and we go to zero to infinity Seeing access to in equals zero So we have to fill it for wild, globalized minus X paradise minus way You called zero to say toe minus Say not that's it in equal to one to infinity. The end plus two into and plus two in the n plus one. Access to an A minus to walk unequaled. Want to infinity, See and and access to and minus sigma. I never would want to infinity seeing access to, and it was a little change and equal to zero. Took unequal smart pulls you toe minus C not plus sigma and a good one to infinity seeing plus two in do and plus two into and plus one minus CNN Doing Linus seeing access to an equals zero. This means to three to minus a nautical zero seeing first to in do and Press two into 10, plus one minus and plus one C and it was zero for and equal to zero. C two equals one by to see note in equal to one. C three was won by three C one, and they go toe to C four. Equals one by four in the one by to see not unequal toe. Three. C four equals one by five. Window one by 3 31 in equal to four. C six equals one by six in the one before in the one by to see not state to and equals one by to restaurant Hindu in factorial in the scene on what he was seeing or Sigma n equal to zero to infinity one day tourist to an Indo in Victoria in the access to to and so you get like sigma any good of zero to infinity expressed when by in factorial equals your restaurants y equals. See not sigma and equal to zero to infinity one by in factorial Hindu extra square by two holders to and which is a co do. So you know, into Here's to half. It's a square y 00 equals zero. See, not equals one. Why? Because it is too X squared by two. That's enough. A question. Thank you.

Today we're going to solve a problem above here. Y dash minus y equals zero. Use my equals. Sigma n equals zero to infinity Seeing extent, it says to sort of So why does equals Sigma n equals one to infinity Seeing star and star access to end minus one by allies equals sigma and equal zero to infinity see and plus one star and plus one express to it Food sigma in equal to zero to infinity See and plus one start and plus one express doing minus sigma a nickel toe one for n equals zero to infinity See and express to n equals zero. So Cigna an equal to zero to infinity seat and plus one and plus one minus c and start express to him for n equals zero See when he calls See, not foreign equals one Steve too. Because half see not which is ego zero. And he goes to C three because one by three star one by to see nautical zero Seeing it was one by in factorial feel Why equal to see you? Yeah, Sigma en equal to zero to infinity. One way in factorial extra student. So light equals C zero are here in states. That's the end off a question. Thank you

We have to solve X double prime minus X times Y prime minus Y equals zero. Given some initial conditions that y zero is one of my primary zero. Yeah, I'm zero. And we're going to use the power series. So our power series expression for Y for Y. Prime for Y double prime, plugging these ball in. Um We get this infinite sequence has to be zero. So we can see here we can shift this um summation to start from zero by shifting N. So we get C seven plus two times and plus two times and plus one times X to the end. This one we can shift the sum to start from zero because um the summation when ana zero is zero and this one we leave alone. So factoring out this ex uh to the end and setting all these coefficients equal to zero, we get C seven plus two equals um and plus one all over and plus two times and plus one C. Saban which is just see Saban divided by M plus two. Now. Um that tells us that C two equals C. Not over to see three equals C 1/3. C four equals C 2/4, which you see not over four times. To C five is C 3/5, which is the 1/5 times three and so on. So we can see that um we can see that, see to the to end right here. These guys is C sub two and equals one over to to the Anton's. In fact, oil temp C not you can also see that if y double prime, if y prime is zero is zero, that tells us that C one has to be zero, which tells us that all the odd odd coefficients have to be zero. No, let's see here, you can plug into our infinite some and we get y equals the summation from zero to infinity. Um Of seen it all over to the N. N. Factorial times X to the two ends. And if why not? If why I'm A away but not in there. Quite zero is one. That means C. One or C not has to be one. So this just goes to one. And then when you can see that this infant some is actually um either the X squared over two. So we've seen that one before. So the solution to this differential equation um with these initial conditions is simply from here, E to the X squared over two. So we have exponential growth, rapid exponential growth.


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