The second order initial value problem below has a power series solution in x. Calculate the first sixcoefficients of the power series, i.e. ao, a₁,... a 5.(3+x²)y"+(5+x)y'—3y= 0, y(0)ao||a1 =a2 =a3||=a4 =a5 ==1, y'(0) = 0

(.) Given initial value problem is,(.) Power series solution of a differential equation about an…

Answered: The second order initial value problem…

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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Related questions

Question

The second order initial value problem below has a power series solution in x. Calculate the first six
coefficients of the power series, i.e. ao, a₁,
... a 5.
(3+x²)y"+(5+x)y'—3y= 0, y(0)
ao
||
a1 =
a2 =
a3
||
=
a4 =
a5 =
=
1, y'(0) = 0

Expert Solution

Step 1: We give standard form of power series solution.

(.) Given initial value problem is,

$open parentheses 3 plus x squared close parentheses y apostrophe apostrophe plus left parenthesis 5 plus x right parenthesis y apostrophe minus 3 y equals 0 space space semicolon space space y left parenthesis 1 right parenthesis equals 0 space comma space y apostrophe left parenthesis 0 right parenthesis equals 0$

(.) Power series solution of a differential equation $y apostrophe apostrophe plus P left parenthesis x right parenthesis y apostrophe plus Q left parenthesis x right parenthesis y equals 0$ about an ordinary point $x equals x subscript 0$ is given by,

$y left parenthesis x right parenthesis space equals space space sum from n equals 0 to infinity of a subscript n open parentheses x minus x subscript 0 close parentheses to the power of n$

(.) $x equals x subscript 0$ is an ordinary point if both $P left parenthesis x right parenthesis$ and $Q left parenthesis x right parenthesis$ are analytic at $x equals x subscript 0$ . i.e. both $P left parenthesis x right parenthesis$ and $Q left parenthesis x right parenthesis$ are defined at $x subscript 0$ .