Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Ammar R.

Differential Equations

1 week, 3 days ago

$x^{2} y^{\prime \prime}-x y^{\prime}+2 y=0 ; \quad x_{0}=2$

Chapter 8

Series Solutions of Differential Equations

Section 4

Equations with Analytic Coefficients

Hello, everyone. Today we're gonna find at least the 1st 4 terms to the general solution to the differential equation X squared. Why? Double prime minus X y Prime plus two y equals zero as a power series about the point eggs, Eero equals two. So the first thing we're gonna do is gonna make the substitution. T equals X minus two So that Xs equal to t plus two An X square is equal to T Square. I was 40 goes for next. We're gonna write why, as a power Siri's of tea. So we get that so some of all and greater than equal to zero of a n t. To the end. The first derivative is this some and greater than equal to one of end some say n time See to the n minus one and the second derivative is some over end greater equal to two of end times and minus one since a m sense t to the n minus two. Now, when you plug this into the equation, we get t square close 40 was four times the sum over and greenery. Quitter, too of end times and minus one times a n teeth of the N minus two minus t plus two. Some over end greater or equal to one of end times a n t to the n minus one plus two. Some were in greenery Code zero of a N T. To the end, which then when we do some expanding, we get some as in greater good, too of end times and minus one terms a NT to the end. That's four some angry nickels to of end since and minus one terms a n t to the n minus one Those four cents to some and green ankle to two of end times and minus one a n t to the n minus two minus the sum and greater anchored to one of end times a n N C to the end minus two times the sum and greater in Quito one of end times a n t to the n minus one plus two tends to some and great American zero of a n t to the end. Now, this looks really on really long and ugly, but we only really care about the 1st 4 terms. So we play in some numbers and we try to get strategically the constant and the coefficients of tea. So the constant terms are to a zero minus to a one. Those 882 and the coefficient of tea is to a one minus 482 minus a one. Those 24 a three plus a two and then the rest we don't care about. So now we get the system to a zero minus to a one, just a two equal zero and a one, because for a two 24 a three equal zero. So this implies that a two is equal to a one minus a zero over four, and a three is equal to a zero minus to a one over 24. So now that's a function of tea. We write a zero one minus 14 T Square was won over 24 t cute. That's whatever toasty one times T plus 1/4 C squared minus won over 12. Thank you, because whatever. And now we substitute t my ex. And so we get a zero one minus 1/4 x minus two square. There's one over 24 x minus two killed. What's wherever plus a one times X minus two plus 1/4 explains to square minus 1 12 X minus two. Cube. That's whatever, and there we have it.

View More Answers From This Book

Find Another Textbook

University of Kansas

Numerade Educator

Other Schools

University of California, Santa Cruz

Villanova University

14:49

$\left(x^{2}-2 x\right) y^{\prime \prime}+2 y=0 ; \quad x_{0}=1$

01:31

$$(x+2)^{2} y^{\prime \prime}-2 y=0$$

01:01

$x^{2} y^{\prime \prime}+2 x^{2} y^{\prime}-2 y=0$

01:12

$$x^{2} y^{\prime \prime}+2 x^{3} y^{\prime}+\left(x^{2}-2\right) y=0$$

01:20

The odds of you getting in…

02:44

Suppose an bag contains 47…

03:04

calculate karl pearson…

02:54

Which of the following is …

01:16

- [ ] 96 students enter a …

01:38

The weight of an aspirin t…

01:59

To determine the average n…

01:13

Marvin's favorite col…

01:04

The function graphed is of…

10:08

Help! Find the inverse Lap…

Enter your parent or guardian’s email address:

Already have an account? Log in

Create an account to get free access

Join Numerade as a

By clicking Sign up you accept Numerade's Terms of Service and Privacy Policy

or sign up with

EMAIL

PASSWORD