Solving a 2nd Order Linear ODE: Any Clever Tricks?

[member 428835]

Homework Statement

$$ay''-(2x+1)y'+2y=0$$ subject to $y(0)=1$ and $y(1)=0$ where $a$ is a non-zero constant.

Homework Equations

Not too sure

The Attempt at a Solution

I know an analytic solution exists since I solved with mathematica. My thoughts were to try a series expansion, but since the analytic solution is in closed form and is using the imaginary error function, I'd rather not waste a lot of time with a power series guess if someone knows (or sees) something insightful.

Any ideas or clever tricks?

Thanks!

 

[HallsofIvy]

I would use two methods. First, let u= 2x+ 1. Then du/dx= 2 so $dy/dx= (dy/du)(du/dx)= 2(dy/du)$ and $d^2y/dx^2= (d/dx)(dy/dx)= d/dx(2dy/du)= 4 d^2y/du^2$. So the equation becomes $4\alpha d^2y/du^2- 2udy/d+ 2y= 0$.

Now look for a power series solution. Let $y= \sum_{n=0}^\infty a_nu^n$.
Content abridged by a mentor.