Some issues for solving differential equations using Fourier transform

Question

Fourier transform is a powerful tool for solving differential equations. But I don't really know when the Fourier transform will give us the full general solution if it can be used. A simple example is the homogeneous simple harmonic oscillator equation: $$\frac{d^2x(t)}{dt^2}+w_0^2x=0$$ If we do a Fourier transform to both sides of the equation, what we will get is actually the trivial solution $x(t)=0$. Another situation is that the same equation with a driving term $f(t)$. In this case we will get some non-trivial solution, but still we will not be able to see the two expected undetermined constants for a second order DE, which means we are still not able to get the full general solution. So could anyone give a general account on using Fourier transform to obtain the general solution?

Answer 1

Actually, that is not the correct conclusion. If one takes the FT of that equation, one gets an equation that may look like

$$\left ( \omega_0^2-\omega^2 \right ) \hat{x} = 0$$

The conclusion to draw is that $\hat{x}=0$ unless $\omega=\omega_0$ or $\omega=-\omega_0$. Thus, the only solutions of the form $e^{i \omega t}$ that are possible are at those frequencies.