Frobenius Method Equal Roots proof

Question

Can someone point me to a proof for the second solution of a DE using Frobenius method when the roots of the indicial equation are equal?

Answer 1

Think of a Cauchy-Euler homogeneous equation $y''+(p/x)y'+(q/x^2)y=0$, where $p$ and $q$ are constants has a solution of the form $y=A.x^r+B.logx.x^r$ in case of equal roots.

Now, the more general form $y''+(\frac{p_0+p_1x+p_2x^2+...}{x}). y'+(\frac{q_0+q_1x+q_2x^2+...}{x^2}).y=0 $ must have a solution of the form $y=A.x^r(a_0+a_1.x+...)+B.logx.x^r(a_0+a_1.x+....)$ from which you can easily see that second solution can be obtained from the first solution by merely multiplying it with $logx$.