(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the terms up to x^5 in the power series solution of the following equation

[tex]y''=(1+x^{2})y[/tex]

2. Relevant equations

Power series, sum from 0 to infinity

[tex]y=\sum a_{n}x^{n}[/tex]

3. The attempt at a solution

At first I just differentiated each term separately and then equated coefficients,

[tex]y=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+a_{4}x^{4}+a_{5}x^{5}+...[/tex]

[tex]y'=a_{1}+2a_{2}x+3a_{3}x^{2}+4a_{4}x^{3}+5a_{5}x^{4}+...[/tex]

[tex]y''=2a_{2}+6a_{3}x+12a_{4}x^{2}+20a_{5}x^{3}+...[/tex]

After equating and rearranging i ended up with,

[tex]y=a_{0}(1+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{12}+\frac{x^{5}}{40}+...)+a_{1}(x+\frac{x^{2}}{2}+\frac{x^{3}}{3}+\frac{x^{4}}{8}+\frac{x^{5}}{24}+...)[/tex]

Is this correct?

Also I was wondering how I would go about doing this for the general case, using complete sums and then equating coefficients,

[tex]y=\sum a_{n}x^{n}[/tex]

[tex]y'=\sum na_{n}x^{n-1}[/tex]

[tex]y''=\sum n(n-1)a_{n}x^{n-2}[/tex]

[tex]x^{2}y=\sum a_{n}x^{n+2}[/tex]

Subbing back into the original,

[tex]\sum n(n-1)a_{n}x^{n-2}=\sum a_{n}x^{n}+\sum a_{n}x^{n+2}[/tex]

Equating coefficients,

[tex]n(n-1)a_{n}=a_{n-2}+a_{n-4}[/tex]

But no matter what I do with this relation I can't get it into a helpful form that agrees with my above result???

Please can someone help?

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# Homework Help: Power series solution to a second order o.d.e.

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