Show that the given solution of the ODE is true. (Show working for the recurrence relationship) 2 dy+さ,32,Ans!ミ+くナ252 Assume asolution of the fo mminy =1,るx"

Given that x3-1y''+x2y'+xy=0 Take y=∑n=0∞cnxn=c0+c1x+c2x2+c3x3+c4x4+c5x5+c6x6+c7c7+. So the…

Answered: 2, +x2 dy Ant ye G(は ) %3B +5(メ+ をナ

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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Show that the given solution of the ODE is true. (Show working for the recurrence relationship)

Assume a
solution of the fo mm
in
y =
1,
るx"

Expert Solution

Step 1

Given that $(x^{3} - 1) y'' + x^{2} y' + x y = 0$

Take

$\begin{array}{rcl} y & = & \sum_{n = 0}^{\infty} c_{n} x^{n} \\ = & c_{0} + c_{1} x + c_{2} x^{2} + c_{3} x^{3} + c_{4} x^{4} + c_{5} x^{5} + c_{6} x^{6} + c_{7} c^{7} + . \end{array}$

So the derivative for y is,

$y' = c_{1} + 2 c_{2} x + 3 c_{3} x^{2} + 4 c_{4} x^{3} + 5 c_{5} x^{4} + 6 c_{6} x^{5} + 7 c_{7} x^{6} + . .$

Take the derivative for y',

$y'' = 2 c_{2} + 6 c_{3} x + 12 c_{4} x^{2} + 20 c_{5} x^{3} + 30 c_{6} x^{4} + 42 c_{7} x^{5} + . .$

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