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I am supposed to find solution of $$xy''+y'+xy=0$$

but i am left with reversing this equation.

i am studying solution of a differential equation by series now and I cannot reverse a series in the form of:

$$ J(x)=1-\frac{1}{x^2} +\frac{3x^4}{32} - \frac{5x^6}{576} ....$$

$$ \frac{1}{J}=1+\frac{x^2}{2} +\frac{5x^4}{32}+ \frac{23x^6}{576}...$$

General formula of $J(x)$ is $$\sum_{n=0}^{\infty} \frac{(-1)n}{(n!)^2}(\frac{x}{2})^2$$

Thanks for all help!!

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# I Solution of an ODE in series Frobenius method

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