2nd-Order (Linear?) Non-Homogeneous ODE, Two Point Boundary Value

[brru25]

Homework Statement

Find the solution to the two-point boundary value problem u'' + 4u' + e^xu = sin(8x)

with u(-1) = u(1) = 0.

Homework Equations

The Attempt at a Solution

I haven't taken an ODE course in years but I need to verify that my numerical solution to the ODE is accurate to the actual solution. The only way to do that is to find the solution to the ODE analytically, which I'm having trouble with. My only guess is to use Variation of Parameters but I can't remember how to use it at all.

 

[Dick]

You can't solve every ode analytically. And yes, the e^x*u term makes it nonlinear. Which probably makes it hard. Just for fun, I plugged into Wolfram Alpha and it gives a completely intractable 'solution' involving definite integrals of 4th order bessel functions. That's not going to be useful to you. You can get a nice, but messy, solution if you leave the e^x out. Can you test your method with that, or maybe something even easier and nicer instead?