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 [itex]dy/dx= (dy/du)(du/dx)= 2(dy/du)[/itex] and [itex]d^2y/dx^2= (d/dx)(dy/dx)= d/dx(2dy/du)= 4 d^2y/du^2[/itex]. So the equation becomes [itex]4\alpha d^2y/du^2- 2udy/d+ 2y= 0[/itex].

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