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## Homework Statement

Hi, So I'm suppose to solve the following problem:

[tex]\left.\frac{d^{2}u}{dt^{2}}-4\frac{d^{3}u}{dt dx^{2}}+3\frac{d^{4}u}{dx^{4}}=0[/tex]

[tex]\left.u(x,0) = f(x)[/tex]

[tex]\left.\frac{du}{dt}(x,0) = g(x)[/tex]

## Homework Equations

## The Attempt at a Solution

First I use fourier transform on the given expression so that I get the following:

Fourier transform of [tex]\left.\frac{d^{2}u}{dt^{2}}(x,t) = \frac{d^{2}\widehat{u}}{dt^{2}}(\omega ,t) [/tex]

Fourier transform of [tex]\left.\frac{du}{dt}(x,t) = \frac{d\widehat{u}}{dt}(\omega ,t) [/tex]

Fourier transform of [tex]\left.\frac{d^{2}u}{dx^{2}}(x,t) = \left(i\omega\right)^{2}\widehat{u}(\omega ,t) = -\left(\omega\right)^{2}\widehat{u}(\omega ,t)[/tex]

Fourier transform of [tex]\left.\frac{d^{4}u}{dx^{2}}(x,t) = \left(i\omega\right)^{4}\widehat{u}(\omega ,t) = \left(\omega\right)^{4}\widehat{u}(\omega ,t)[/tex]

Which means me overall expression after transform is:

[tex]\left.\frac{d^{2}\widehat{u}}{dt^{2}}(\omega ,t)+4\left(\omega\right)^{2}\frac{d\widehat{u}}{dt}(\omega ,t)+3\left(\omega\right)^{4}\widehat{u}(\omega ,t)=0[/tex]

Now assuming I did that correctly, the next step I think I should proceed with is to solve for [tex]\left.\widehat{u}(\omega ,t)[/tex]. I don't remember how to solve this type of ODE, I was reading a couple of sites and it says I should use a characteristic equation which would I assume then be, [tex]\left.\lambda^{2}+4\omega^{2} \lambda +3\omega[/tex] where [tex]\lambda[/tex] is just an arbitrary symbol to denote a quadratic equation. I looked for the roots and used it along with the general expression of the 2nd order ODE to get

[tex]\left.\widehat{u}(\omega ,t)=c_{1}+c_{2}e^{-4\omega^{2}t}[/tex]

But it seems to be incorrect since I took the derivative and plugged it back into my fourier transform expression and did not get a 0 for my answer so...Any guidance would be much appreciated!!! Thanks!