Given the differential equation:(1-x)y"+y=0, x0=0Find:Seek the power series solution for the differential equation about the given point x0; find the recurrence relationFind the first four terms in each of the two solutions, y1 and y2 (unless the series terminates sooner)By evaluating the Wronskian, W(y1,y2)(x0), show that y1 and y2 form a fundamental set of solutionsIf possible, find the general term in each solution

Question
Asked Jul 2, 2019

Given the differential equation:

(1-x)y"+y=0, x0=0

Find:

  • Seek the power series solution for the differential equation about the given point x0; find the recurrence relation
  • Find the first four terms in each of the two solutions, y1 and y2 (unless the series terminates sooner)
  • By evaluating the Wronskian, W(y1,y2)(x0), show that y1 and y2 form a fundamental set of solutions
  • If possible, find the general term in each solution
check_circleExpert Solution
Step 1

Consider the differential equation

(1-х) у" + у -0, х, 3D0.
Let a power series solution of the differential solution be
y e(x-) that is y e
as x 0
n 0
n-0
and y" (n-c
Then y >n
п-1
пс, х
n-1
n-2
help_outline

Image Transcriptionclose

(1-х) у" + у -0, х, 3D0. Let a power series solution of the differential solution be y e(x-) that is y e as x 0 n 0 n-0 and y" (n-c Then y >n п-1 пс, х n-1 n-2

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Step 2

Substitute y, y’ and y’’ in

(1-x)y"y0
n(n-1)cx e" |
(1-
0
x
n-2
n=0
Žn(n-1)e,x ]+£€,x".
n(n-1)c
37
=0
C
n=2
n-2
n(n-l)c (n-1)e,x
0
n-2
n-0
+
_
n+1
n-0
n-1
n=0
help_outline

Image Transcriptionclose

(1-x)y"y0 n(n-1)cx e" | (1- 0 x n-2 n=0 Žn(n-1)e,x ]+£€,x". n(n-1)c 37 =0 C n=2 n-2 n(n-l)c (n-1)e,x 0 n-2 n-0 + _ n+1 n-0 n-1 n=0

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Step 3

Thus, the two equation...

co
Σ.
2c, (n+2) (n+1)
n+1)(n)c
= 0
Cr
n+1
n-1
n-1
n-1
2c, + c0 or(n+2)(n +1)e -(n +1) (n) cm" £" = 0
n-1
n=1
n=1
The value of c2 becomes c
2
help_outline

Image Transcriptionclose

co Σ. 2c, (n+2) (n+1) n+1)(n)c = 0 Cr n+1 n-1 n-1 n-1 2c, + c0 or(n+2)(n +1)e -(n +1) (n) cm" £" = 0 n-1 n=1 n=1 The value of c2 becomes c 2

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