# problem with recurrence relation for series solution for ODE

I have $$y''-xy'-y=0$$ and I'm trying to find the series solution around the ordinary point $x_0=1$. My last post I muscled through to the solution when the ordinary point was $x_0=0$, but this is proving to be tougher. Now I have obtained through power series analysis $$y''-xy'-y=0=\sum_{k=0}^{\infty}[(k+2)(k+1)a_{k+2}-(k+1)a_{k+1}-(k+1)a_k](x-1)^k$$ which yields the recurrence relation $$a_{k+2}=\frac{a_{k+1}+a_{k}}{k+2}$$ WHen I start plugging in sequential "k" values I'm not finding a very good pattern emerging. $$a_2=\frac{a_0}{2!}+\frac{a_1}{2!}$$ $$a_3=\frac{a_0}{3!}+\frac{3a_1}{3!}$$ $$a_4=\frac{4a_0}{4!}+\frac{6a_1}{4!}$$ $$a_5=\frac{8a_0}{5!}+\frac{18a_1}{5!}$$ $$a_6=\frac{28a_0}{6!}+\frac{48a_1}{6!}$$ $$a_7=\frac{76a_0}{7!}+\frac{156a_1}{7!}$$ Outside just writing out term by term, substituting in the appropriate $a_k$, does this have a nice closed form? Or is is just the case that I have the answer with the $a_k$'s that I have

• Generally speaking, what you have already done (getting a recursive form for the coefficients) is the best you can do when finding power series solutions. – JohnD Nov 29 '13 at 4:44
• I checked your recurrence relation and it is correct. – Amzoti Nov 29 '13 at 5:19

$$y=e^{\frac{x^2}{2}} \left(C_2+C_1\sqrt{\frac{\pi }{2}} \text{erf}\left(\frac{x}{\sqrt{2}}\right)\right)$$
As you see, you have two arbitrary constants and then any polynomial expansion of the solution will depend on two constants : these are your $$a_0$$ and $$a_1$$ terms.