Answered: y(1) =2 愛+y=0; 3 y') =4. 24 4(メー)-4(メー)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

By using the method of substitution, where t=x-1, show that the power series solution in powers of (x-1) given is true.

y(1) =2
y') =4,
244(メー)-4(メー) -
メ
dメ
Ansi
+ 2(メー)1T+

Expert Solution

Step 1

Substituting $y = \sum_{n = 0}^{\infty} c_{n} {(x - 1)}^{n}$ in $x \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} + 2 y = 0$ :

$x \frac{d^{2} y}{d x^{2}} + \frac{d y}{d x} + 2 y = 0 x \sum_{n = 2}^{\infty} n (n - 1) c_{n} {(x - 1)}^{n - 2} + \sum_{n = 1}^{\infty} n c_{n} {(x - 1)}^{n - 1} + 2 \sum_{n = 0}^{\infty} c_{n} {(x - 1)}^{n} = 0 (x - 1) \sum_{n = 2}^{\infty} n (n - 1) c_{n} {(x - 1)}^{n - 2} + \sum_{n = 2}^{\infty} n (n - 1) c_{n} {(x - 1)}^{n - 2} + \sum_{n = 1}^{\infty} n c_{n} {(x - 1)}^{n - 1} + 2 \sum_{n = 0}^{\infty} c_{n} {(x - 1)}^{n} = 0 \sum_{n = 2}^{\infty} n (n - 1) c_{n} {(x - 1)}^{n - 1} + \sum_{n = 2}^{\infty} n (n - 1) c_{n} {(x - 1)}^{n - 2} + \sum_{n = 1}^{\infty} n c_{n} {(x - 1)}^{n - 1} + 2 \sum_{n = 0}^{\infty} c_{n} {(x - 1)}^{n} = 0 \sum_{n = 1}^{\infty} n (n + 1) c_{n + 1} {(x - 1)}^{n} + \sum_{n = 0}^{\infty} (n + 1) (n + 2) c_{n + 2} {(x - 1)}^{n} + \sum_{n = 1}^{\infty} (n + 1) c_{n + 1} {(x - 1)}^{n} + 2 \sum_{n = 0}^{\infty} c_{n} {(x - 1)}^{n} = 0$

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