Differential equations by series and also by an elementary method

[Fachni]

Homework Statement

Solve the following differential equations by series and also by an elementary method and verify that your solutions agree.

$$(x^2+2x)y''-2(x+1)y'+2y=0$$

Homework Equations

$$y=\sum_{n=0}^{\infty} a_nx^n$$
$$y'=\sum_{n=1}^{\infty} na_nx^{n-1}$$
$$y''=\sum_{n=1}^{\infty} n(n+1)a_{n+1}x^{n-1}$$

The Attempt at a Solution

I have got $$a_1=a_0,\ a_3=0,\ a_4=-\frac{1}{8}a_3=0,\ a_5=-\frac{1}{5}a_4=0$$. Then, how do we find the recursive relation to find the solution?

 

[vela]

Isn't x=0 a singular point? You need to use the Frobenius method to get the series solution.

For the recursion relation, set the coefficient of xⁿ to 0.