- #1

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## Main Question or Discussion Point

I just finished a final in my differential equations class. One of the problems had me solve a second order homogeneous differential equation using series. I boiled it down to this recursion relation:

[tex]a_{n+2}=\frac{(n+3)a_{n}}{2(n+2)(n+1)}[/tex]

I found that the even coefficients work out nicely to the following sum:

[tex]y=a_{0}+\Sigma^{\infty}_{n=2}\frac{(n+1)a_{0}}{4*6^{n-2}}x^{n}[/tex]

I couldn't get a nice result for the odd coefficients and still can't find one. It's kind of bothering me now. Is it even possible? I can boil it down to this series:

[tex]\frac{1}{3},\frac{1}{20},\frac{1}{210},\frac{1}{3024},\frac{1}{55440},...[/tex]

[tex]a_{n+2}=\frac{(n+3)a_{n}}{2(n+2)(n+1)}[/tex]

I found that the even coefficients work out nicely to the following sum:

[tex]y=a_{0}+\Sigma^{\infty}_{n=2}\frac{(n+1)a_{0}}{4*6^{n-2}}x^{n}[/tex]

I couldn't get a nice result for the odd coefficients and still can't find one. It's kind of bothering me now. Is it even possible? I can boil it down to this series:

[tex]\frac{1}{3},\frac{1}{20},\frac{1}{210},\frac{1}{3024},\frac{1}{55440},...[/tex]