# Second-order nonhomogeneous diff-eq

BucketOfFish
Hey guys. It's been a few years since I've taken diff-eq and I can't remember how to solve second-order problems like this one:

Find the general solution of
u'' + a^2*u = cos(bx)

I know that if it were homogeneous, I would solve for r^2 + a^2 = 0, and get u = ce^(rx). But for the life of me I can't remember what to do with that cosine on the right side of the equation. Can anyone help?

Homework Helper
Welcome to PF!

Hi BucketOfFish! Welcome to PF! (try using the X2 icon just above the Reply box )

You need to find a particular solution (ie any solution to the equation), which you can then add to the general solution to the homogeneous equation.

In this case, try a combination of cos(bx) and sin(bx). (if b = ±a, that doesn't work, and you'll need xcos(bx) and xsin(bx))

BucketOfFish
Thanks a lot; that really helped!

I bet you could also use the function:

u''+a2u' = eibx and split the real and imaginary portions with Euler's formula.

I find e easier to work with than sin and cos.

hi evad1089! even when cos is already on the RHS? 