1. ## Fourier transforms

Use Fourier transforms to solve the ODE
-uxx + (a^2)u = δ(x)

2. Originally Posted by jin_nzzang
Use Fourier transforms to solve the ODE
-uxx + (a^2)u = δ(x)

Well you could start by taking the FT of this.

If we take the definition:

$F(\omega)=\mathcal{F}f (\omega)=\frac{1}{\sqrt{2\pi}}\int_{-infty}^{\infty}f(x) e^{-i\omega t}\; dt$

then we have:

$
(\mathcal{F}D^nf) (\omega)=(i\omega)^n F(\omega)
$

and $\mathcal{F} \delta(\omega)=\frac{1}{\sqrt{2\pi}}$

Then the equation becomes after taking the FT:

$
-(i\omega)^2U(\omega)+a^2U(\omega)=\frac{1}{\sqrt{2 \pi}}
$

Now solve this for $U(\omega)$ and take the inverse FT

CB