Answered: (1+x2)y''+2xy'-2y=0 y(0)=1, y'(0)=1…

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

(1+x²)y''+2xy'-2y=0 y(0)=1, y'(0)=1

What is the power series solution about x=0

Expert Solution

Step 1: Power series derivation

Given ODE is $open parentheses 1 plus x squared close parentheses y apostrophe apostrophe plus 2 x y apostrophe minus 2 y equals 0 comma y left parenthesis 0 right parenthesis equals 1 comma y apostrophe left parenthesis 0 right parenthesis equals 1$

Observed that here $x equals 0$ is ordinary point for given ODE,So it's power series solution is of the form about $x equals 0$

$y left parenthesis x right parenthesis equals sum from n equals 0 to infinity of a subscript n x to the power of n rightwards double arrow y apostrophe left parenthesis x right parenthesis equals stack sum n with n equals 0 below and infinity on top a subscript n x to the power of n minus 1 end exponent space a n d space y apostrophe apostrophe left parenthesis x right parenthesis equals stack sum n with n equals 0 below and infinity on top left parenthesis n minus 1 right parenthesis a subscript n x to the power of n minus 2 end exponent space$

Now by substituting we get $open parentheses 1 plus x squared close parentheses sum from n equals 0 to infinity of n left parenthesis n minus 1 right parenthesis a subscript n x to the power of n minus 2 end exponent plus 2 x sum from n equals 0 to infinity of n a subscript n x to the power of n minus 1 end exponent minus 2 sum from n equals 0 to infinity of a subscript n x to the power of n equals 0$

$rightwards double arrow sum from n equals 2 to infinity of n left parenthesis n minus 1 right parenthesis a subscript n x to the power of n minus 2 end exponent plus sum from n equals 2 to infinity of n left parenthesis n minus 1 right parenthesis a subscript n x to the power of n plus sum from n equals 0 to infinity of 2 n a subscript n x to the power of n minus sum from n equals 0 to infinity of a subscript n x to the power of n equals 0$

$rightwards double arrow sum from n equals 0 to infinity of open parentheses n plus 2 close parentheses left parenthesis n plus 1 right parenthesis a subscript n plus 2 end subscript x to the power of n plus sum from n equals 0 to infinity of open parentheses n plus 2 close parentheses left parenthesis n plus 1 right parenthesis a subscript n plus 2 end subscript x to the power of n plus 2 end exponent plus sum from n equals 0 to infinity of 2 open parentheses n plus 1 close parentheses a subscript n plus 1 end subscript x to the power of n plus 1 end exponent minus sum from n equals 0 to infinity of a subscript n x to the power of n equals 0$

$rightwards double arrow sum from n equals 0 to infinity of open parentheses n plus 2 close parentheses open parentheses n plus 1 close parentheses a subscript n plus 2 end subscript x to the power of n plus 2 end exponent plus sum from n equals 0 to infinity of 2 open parentheses n plus 1 close parentheses a subscript n plus 1 end subscript x to the power of n plus 1 end exponent plus sum from n equals 0 to infinity of open square brackets open parentheses n plus 2 close parentheses open parentheses n plus 1 close parentheses a subscript n plus 2 end subscript minus a subscript n close square brackets x to the power of n equals 0$