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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 $\alpha \cdot \beta$.

  • 1
    What have you tried? This is very similar to [this other question you just posted](http://math.stackexchange.com/questions/255974/how-to-find-the-general-solution-of-1x2y2xy-2y-0-how-to-express-by-me), which you seemed to be able to get started on.2012-12-11
  • 0
    I'm mainly not sure what it's asking.2012-12-11

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