Question

Due Tomorrow, and it's 1:30 in the morning.

Given the Linear ODE y''-xy'+2y=0
(a). Find two linearly independent power series solutions of this equation.
(b). What is the radius of convergence of these series?

Answer 1

\[y(x)=\sum_{n=0}^\infty a_n x^n \\ y'(x)=\sum_{n=1}^\infty n a_n x^{n-1} \\ y''(x)=\sum_{n=2}^\infty n(n-1)a_nx^{n-2} \\ \sum_{n=2}^\infty n(n-1)a_nx^{n-2}-x \sum_{n=1}^{\infty} na_nx^{n-1}+2 \sum_{n=0}^{\infty} a_nx^n=0 \\ \\ \sum_{n=0}^\infty (n+2)(n+1)a_{n+2}x^{n}- \sum_{n=1}^{\infty}na_{n}x^{n}+2 \sum_{n=0}^\infty a_nx^n=0 \\ \\ \\ \]
\[2(1)a_2+2a_0+\sum_{n=1}^\infty[ (n+2)(n+1)a_{n+2}x^{n}-na_nx^{n}+2a_nx^{n}]=0 \\ \\ 2a_2+2a_0+\sum_{n=1}^{\infty} x^n[(n+2)(n+1)a_{n+2}-n a_n+2 a_n]=0 \\ \]
maybe you can going from here