Expressing this series in closed form Watch

wanderlust.xx
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#1
Report Thread starter 7 years ago
#1
So I'm finding a series solution to a DE, and I got the following relation:

a_n = \dfrac{a_{n-1}}{n(2n-1)} = \dfrac{2a_{n-1}}{2n(2n-1)}

Repeating this, I found

a_n = \dfrac{2^n a_0}{(2n)!}

So my solution in this case must be

y(x) = \displaystyle\sum_{n=1}^{\infty} \dfrac{2^n a_0}{(2n)!}x^n = a_0 \displaystyle\sum_{n=1}^{\infty} \dfrac{(\sqrt{2})^{2n} }{(2n)!}(\sqrt{x})^{2n} =  a_0 \displaystyle\sum_{n=1}^{\infty} \dfrac{(\sqrt{2x})^{2n})}{(2n)!}  = a_0 \ cosh {\sqrt{2x}}

I believe.

I managed to get \dfrac{a_0}{\sqrt{2}} sinh(\sqrt{2x}) as my second solution. Does this seem... valid?

Nevermind, wolfram says it's cool.
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