Driven, damped harmonic oscillator - need help with particular solution

1. Nov 6, 2005

eku_girl83

Driven, damped harmonic oscillator -- need help with particular solution

Consider a damped oscillator with Beta = w/4 driven by
F=A1cos(wt)+A2cos(3wt). Find x(t).

I know that x(t) is the solution to the system with the above drive force.

I know that if an external driving force applied to the oscillator then the total force is described by F = -kx - bx' + F0cos(wt).

But in our case the driving force is A1cos(wt)+A2cos(3wt) so
F=-kx-bx+A1cos(wt)+A2cos(3wt).

Then our differential equation is mx''+bx'+kx=A1cos(wt)+A2cos(3wt).

This can also be written as x''+2Betax'+(w^2)x=A1cos(wt)+A2cos(3wt).

For the complementary solution, we set the right side of the equation equal to zero and solve for x. This is o.k.

However, I am having trouble with the particular solution. Can someone tell me how I find a particular solution for this? I can find the particular solution for x''+2Beta x'+ (w^2)x = A cos (wt), but what about the particular solution when the driving force is not A cos (wt), as we have in this case?

Any help GREATLY appreciated!

2. Nov 7, 2005

dextercioby

Use Lagrange's method of varying constants. That is assume that the particular solution to the nonhomogenous ODE is

$$x_{part}(t)=C_{1}(t)\cos\omega t+C_{2}(t)\cos 3\omega t$$.

Daniel.