Second-order nonhomogeneous ODE with undetermined coefficients

[andrewdavid]

(d^2x/dt^2)+(w^2)x=Fsin(wt), x(0)=0,x'(0)=0

Hope that's readable. First part is second derivative of x with respect to t. w is a constant and F is a constant. I need to find a solution to this using method of undetermined coeffecients and I'm confused with all the different variables. Anyone get me started at least?

 

[Pyrrhus]

Well, first off start by solving the homogenous equation to find the fundamental solution.

$$\ddot{x} + \omega^{2}x = 0$$

After that try a Particular solution of the type

$$y_{p} = A x \sin(\omega t) + B x\cos(\omega t)$$

Remember that if the fundamental solution has already sin and cos, you will need to try a xsin and xcos, like this case.

 

[andrewdavid]

I got my homogenous equation x''+(w^2)x=0 but I can't find my roots with that w^2 in there.

 

[Pyrrhus]

What seems to be the problem? Show me your work.

 

[Pyrrhus]

Here, i will start you off

$$\ddot{x} + \omega^{2}x = 0$$

we assume a as a solution

$$x(t) = e^{rt}$$

So we substitute in our ODE

$$r^{2}e^{rt} + \omega^{2}e^{rt} = 0$$

so

$$e^{rt}(r^{2} + \omega^{2}) = 0$$

because $e^{rt}$ cannot be equal to 0

$$r^{2} + \omega^{2} = 0$$

which ends up as

$$r = \pm \omega i$$

 

[andrewdavid]

I figured it out, thanks a lot for your help, I was just being dumb.