explicit solution for second order homogeneous linear differential equation If we consider the equation

$(1-x^2)\dfrac{d^2y}{dx^2} -2x \dfrac{dy}{dx}+2y=0, \quad -1<x<1$

how can we find the explicit solution, what should be the method for solution?

There is one obvious particular solution which is $y=c_1 x$. The second one is much less obvious to me but, using a CAS, I found as general solution $$y=c_1 x+c_2 \left(x \tanh ^{-1}(x)-1\right)$$ I hope and wish this will give you some ideas. As said by achille hui, beside the solution in terms of Legendre polynomials, you can obviously use Frobenius method which leads to the solution I wrote.
