How do I make the power series of $4x^2y''(x)-4x^2y'(x) + (1-2x)y(x)=O$? I want to do this with the frobenius method, considering me the first solution is $\sum_{k=0}^{\infty}1/k! * x^{k+1/2}$. I'm quite sure this one is correct. But when I try to make the second solution I don't get to it.
Thanks in advance.
It is $$y \left( x \right) ={\it \_C1}\,\sqrt {x} \left( (1+x+{\frac{1}{2}}{x} ^{2}+{\frac{1}{6}}{x}^{3}+{\frac{1}{24}}{x}^{4}+{\frac{1}{120}}{x}^{5} +O \left( {x}^{6} \right) ) \right) +{\it \_C2}\, \left( \sqrt {x}\ln \left( x \right) \left( (1+x+{\frac{1}{2}}{x}^{2}+{\frac{1}{6}}{x}^{ 3}+{\frac{1}{24}}{x}^{4}+{\frac{1}{120}}{x}^{5}+O \left( {x}^{6} \right) ) \right) +\sqrt {x} \left( (-x-{\frac{3}{4}}{x}^{2}-{\frac{ 11}{36}}{x}^{3}-{\frac{25}{288}}{x}^{4}-{\frac{137}{7200}}{x}^{5}+O \left( {x}^{6} \right) ) \right) \right) $$