Verify that the equation $y''+y'-xy=0$ has a three-term, recursion formula and find its series solutions $y_1$ and $y_2$ such that
$a)$ $y_1(0)=1$, $y_1'(0)=0$;
$b)$ $y_2(0)=0$, $y_2'(0)=1$.
Using the following theorem, of which it guarantees that both series converge at ever $x\in \mathbb{R}$, let $\displaystyle\sum_{j=0}^{ \infty } a_j$ and $\displaystyle\sum_{j=0}^{ \infty } b_j$ be two absolutely converfent series which converge to limits $\alpha$ and $\beta$, respectively. Define the seies as $\displaystyle\sum_{j=0}^{ \infty } c_m$ with summands $c_m=\displaystyle\sum_{n=0}^{ m } a_j\cdot b_{m-j}$. Then the series $\displaystyle\sum_{m=0}^{ \infty } c_m$ converges to $\alpha \cdot \beta$.