# Oddly Formatted Second Order ODE

1. Oct 20, 2014

### checkmatechamp

1. The problem statement, all variables and given/known data
u'' + w20*u = cos(wt)

w refers to omega.

2. Relevant equations

3. The attempt at a solution

I'm not sure where to begin on this. For starters, it's a multiple choice problem, and all the answers are given in terms of y, so I'm not sure if u is supposed to replace y' or something.

Second of all, even if I start solving it, trying to use r2 + 1*w20 = 0, and then getting that r = w0, where do I go from there? I have a general solution of y = c1ew0t + c2e-w0t

So then I say that the particular solution has to have the form A*cos(wt) + B*sin(wt), and the second derivative is -A*cos(wt) - B*sin(wt)

So then A*cos(wt) - B*sin(wt) + w02*A*cos(wt) + B*sin(wt) = cos(wt)

So then A*cos(wt) + w02*A*cos(wt) = 1*cos(wt)

So then that gets me that A + w02*A = 1, and A = 1/(w02)

The sine terms cancel (and there's no sine in the solution anyway)

So then I have y = c1ew0t + c2e-w0t + (cos(wt))/(w02).

The closest choice I see is c1*cos(w0t) + c2*sin(w0t) + (cos(wt))/w02, and that choice is wrong.

2. Oct 20, 2014

### vela

Staff Emeritus
$r=\omega_0$ isn't a solution. Plugging it into the characteristic equation would give you $2\omega_0^2 = 0$, which isn't true in general.

You didn't differentiate correctly, you dropped a sign, and you made an algebra errors distributing $\omega_0^2$.