Series Solutions

Facts

Existence of Power Series Solutions

Let $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=0⊛$ be a 2nd order ODE.

If $x_0$ is an ordinary point of the ODE, then Every solution of the ODE is analytic at $x_0\iff R>0$ s.t. $y(x)=\sum _{n=0}^{\infty }{a}_n(x-x_0{)}^n$ on $|x|<R$

$y(x)=a_0{y}_1(x)+a_1{y}_2(x)$, where $y_1,y_2$ are solutions with $y_1(x_0)=1,y_1^{\prime }(x_0)=0,$ such $\{y_1,y_2\}$ is basis

The radius of convergence $R$ of $y(x)$ is at least the minimum of those of $p(x),q(x)$

Solution

Transclude of Frobenius-Method