MHB How to solve an ODE in powers of x?

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I have this question due soon and I have no idea how to do it. Please help me get started on it

Solve in powers of x: (1-x^2)y''-2xy'+42y=0
 
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Re: Solving in powers

The equation:

$$\left(1-x^2\right)y''-2xy'+n(n+1)y=0$$

where $n\in\mathbb{N}$ is called Legendre's differential equation which has as solutions:

Legendre polynomials

That should give you a place to start. :D
 
Thank You, Mark! I'll try working on it and reply with any problems I have
 

Thread 'Volterra equation of the second kind'

There is the following linear Volterra equation of the second kind $$ y(x)+\int_{0}^{x} K(x-s) y(s)\,{\rm d}s = 1 $$ with kernel $$ K(x-s) = 1 - 4 \sum_{n=1}^{\infty} \dfrac{1}{\lambda_n^2} e^{-\beta \lambda_n^2 (x-s)} $$ where $y(0)=1$, $\beta>0$ and $\lambda_n$ is the $n$-th positive root of the equation $J_0(x)=0$ (here $n$ is a natural number that numbers these positive roots in the order of increasing their values), $J_0(x)$ is the Bessel function of the first kind of zero order. I...
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