# Power Series Solution of Second Order DE

$$y''+y'+xy'-y=0$$

-
y"+y'+(x)(y')-y=0 –  killerAk40 Dec 15 '13 at 19:46
Where do you want centered the power series ? –  Tony Piccolo Dec 15 '13 at 23:03

Let $y(x)=\sum_{i=0}^{\infty}a_i x^i$. Find $y'$ and $y''$ by differentiating $y(x)$.
$$y'(x)=\sum_{i=0}^{\infty}ia_i x^{i-1}$$
$$y''(x)=\sum_{i=0}^{\infty}i(i-1)a_i x^{i-2}$$
Substitute all these in the differential equation and try to find a recurrence relation for $a_i$. Hope you can continue. You can find an example of the power series method >>here<<.