Introducing homogeneous solution into Fourier Analysis of differential equation

2018-07-22 02:52:04

I am interested in solving the following system using Fourier Analysis because I'm trying to show how this particular system acts as a filter bank:

$$ \frac{dp(t)}{dt} = k s(t) - \beta p(t) $$

where $s(t)$ is some driving force. This is essentially the equation of the velocity of a driven, damped oscillator.

Doing Fourier Analysis on this equation I get that:

$$ (i\omega + \beta) P(\omega) = k S(\omega) $$

Now, the particular solution is given by dividing both sides by $(i\omega + \beta)$, but to get the general solution, the solution to the homogenous equation must be included as well. This is the solution of the equation $(i\omega+ \beta)P(\omega) = 0$. I deduce that the solution of that equation is $C \delta(\omega - i\beta)$. Thus the solution in Fourier Space of all this is

$$ P(\omega) = \frac{k}{i\omega + \beta} S(\omega) + C \delta(\omega - i\beta). $$

The only weird thing about this is that when I do the inverse Fourier transform, I don't think the con