Trying to find dispersion relation

[autobot.d]

Homework Statement

$\imath\frac{\partial u}{\partial t} + \frac{\partial^2 u}{\partial x^2}=0$

$\left(x,t\right) = \int^{\infty}_{-\infty}A\left(k\right)e^{\imath\left(kx-wt\right)}dk$

$u\left(x,0\right)=\delta\left(x\right)$

Homework Equations

Not sure how to get w(k)

The Attempt at a Solution

$A\left(k\right) = \frac{1}{2\pi}\int^{\infty}_{-\infty}\delta\left(x\right)e^{-\imath\left(kx\right)}dx = \frac{1}{2\pi}$

plugging this into u(x,t) do I work with


$u\left(x,t\right) =\frac{1}{2\pi}\int^{\infty}_{-\infty}e^{\imath\left(kx-wt\right)}dk$

This is where I am stuck. I know w(k) is the dispersion relation. If I put in the pde do I just deal with

$\imath \left(-\imath w\right) + \frac{d^{2}u}{dt^{2}} = w +\frac{d^{2}u}{dt^{2}}=0$

?

Not sure if this is what I even want to do. Any guidance would be appreciated.

 

[autobot.d]

I was making the problem too hard on myself. I got

$w=k^2$