# Frobenius Method - Non integer powers of $x$ in differential equation?

I am trying to solve an ODE using the Frobenius method. I understand the general process, but I do not understand how you compare coefficients when you have a $x^\frac{1}{2}$ term in the differential equatio. All my searches on google just go to non-integer differences in the indicial equation.

For example an equation such as

$x^2y''+(\sqrt{x}-K)y = 0 \\$

using the Frobenius series

$y = \sum_{n=0}^\infty a_n x^{n+s}$

Substituting this and its derivatives into the differential equation

$\sum_{n=0}^\infty a_n x^{n+s+2}+\sum_{n=0}^\infty a_n x^{n+s+1/2}-K\sum_{n=0}^\infty a_n x^{n+s} =0$

Normally I would make the first term start from $n=2$ or make $a_n\rightarrow a_{n-2}$ and do the same for the remaining terms to be able to compare powers of x.

How do you proceed when there is a non-integer power of x in the differential Equation?

• Substitute $x^{1/2}=\alpha$ and calculate the method for the equation $$\alpha^4y''(\alpha^2)+(\alpha-K)y'(\alpha^2)=0.$$ – ty. Feb 17 '18 at 13:10
• thanks @ty. In this case does my frobenius series look like $y(\alpha^2)=\sum_{n=0}^\infty a_n\alpha^{2n+s}$ – johnahh Feb 17 '18 at 14:02