How to solve this second order equation using power series method?

**redhotman225** · November 26th 2012, 07:03 PM

Find the power series solutions of the equation:

(2x²)y'' + (3x)y' - (x² + 1)y = 0

Any help would gladly be appreciated.

**Prove It** · November 26th 2012, 08:00 PM

Start by writing $\displaystyle \begin{align*} y = \sum_{ k = 0 }^{\infty} c_k \, x^k \end{align*}$ , which means $\displaystyle \begin{align*} y' = k\, c_k \, x^{k-1} \end{align*}$ and $\displaystyle \begin{align*} y'' = k\left( k - 1 \right) c_k \, x^{k - 2} \end{align*}$ .

Substitute these in and perform some algebraic manipulation. What can you come up with?

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