Frobenius Method of Solving ODE

[IImattII]

Homework Statement

Find two series solutions of the given DE about the regular point x = 0.
2xy'' + 5y' + xy = 0

Homework Equations

The answer is in the form y = SUM(c_nx^n+r) where SUM is the sum from n = 0 to infinity.

The Attempt at a Solution

This questions got me stumped. I'm able to substitute the second and first order derivatives into the equation. I'm also able to to combine the different sums so they start at the same index. The equation I end up with is:

x^r[r(2r+3)c₀x^-1 + [2r(r+1)+5(r+1)]c₁ + SUM[(k+r+1)(2k+2r+5)c_k+1 + c_k-1]x^k] = 0
where in this case, SUM is the sum from k = 1 to infinity

I guess my problem is determining the indicial equation and thus the recursive formulas. I've tried setting the coefficients of the x^-1 term equal to 0 and using that as my indicial equation resulting in r = 0 and -3/2 but the two resulting recursive formulas end in a solution with two series each so instead of only two series solutions I have four.

 

[lippyka]

I'm assuming I'm just not setting up the problem correctly. Any help/suggestions would be greatly appreciated!