- #1

roldy

- 237

- 2

## Homework Statement

Consider the partial differential equation

[tex] u_{xx}-3u_{xt}-4u_{tt}=0 [/tex]

(a) Find the general solution of the partial differential equation in the xt-plane, if possible.

(b) Find the solution of the partial differential equation that satisfies

[tex]u(x,0)=x^3[/tex] and [tex]u_t(x,0)=-3x^2[/tex] for [tex]-\infty<x<\infty[/tex]

## Homework Equations

## The Attempt at a Solution

(a)

[tex]

\left(\frac{\partial^2}{\partial x^2}-\frac{3\partial^2}{\partial x\partial t}-\frac{4\partial^2}{\partial t^2}\right)u=0

[/tex]

*

[tex]

\left(\frac{\partial}{\partial x}+\frac{\partial}{\partial t}\right)\left(\frac{\partial}{\partial x}-\frac{4\partial}{\partial t}\right)u=0

[/tex]

Set the coefficients equal to a variable...

[tex]\alpha=1, \beta=1, \gamma=1, \delta=-4[/tex]

[tex]

\zeta=\beta{x}-\alpha{t}=x-t

[/tex]

[tex]

\eta=\delta{x}-\gamma{t}=-4x-t

[/tex]

[tex]

\frac{\partial v}{\partial x}=\frac{\partial v}{\partial \zeta}\frac{\partial \zeta}{\partial x}+\frac{\partial v}{\partial \eta}\frac{\partial \eta}{\partial x}=\frac{\partial v}{\partial \zeta}-\frac{4\partial v}{\partial \eta}

[/tex]

[tex]

\frac{\partial v}{\partial t}=\frac{\partial v}{\partial \zeta}\frac{\partial \zeta}{\partial t}+\frac{\partial v}{\partial \eta}\frac{\partial \eta}{\partial t}=\frac{-\partial v}{\partial \zeta}-\frac{\partial v}{\partial \eta}

[/tex]

So now substitute these two differentials into *

[tex]

\left[\left(\frac{\partial}{\partial \zeta}-\frac{4\partial}{\partial \eta}\right)+\left(\frac{-\partial}{\partial \zeta}-\frac{\partial}{\partial \eta}\right)\right]\left[\left(\frac{\partial}{\partial \zeta}-\frac{4\partial}{\partial \eta}\right)-\left(\frac{-4\partial}{\partial \zeta}-\frac{4\partial}{\partial \eta}\right)\right]u=0

[/tex]

[tex]

\left(\frac{-4\partial}{\partial \eta}-\frac{\partial}{\partial \eta}\right)\left(\frac{\partial}{\partial \zeta}+\frac{4\partial}{\partial \zeta}\right)u=0

[/tex]

[tex]

\left(\frac{-5\partial}{\partial \eta}\right)\left(\frac{5\partial}{\partial \zeta}\right)u=0

[/tex]

[tex]

\frac{\partial}{\partial \eta}\left(\frac{\partial u}{\partial \zeta}\right)=0

[/tex]

Let [tex]h(\zeta)=\frac{\partial u}{\partial \zeta}[/tex]

So now integrate with respect to [tex]\zeta[/tex] holding [tex]\eta[/tex] fixed.

[tex]

u=\int h(\zeta)d\zeta + g(\eta)=f(\zeta)+g(\eta)

[/tex]

Substitute back in for [tex]\zeta[/tex] and [tex]\eta[/tex]...

[tex]

u(x,t)=f(x-t)+g(-4x-t)

[/tex]

Now this should be the general solution. Is this right, did I do my differentiation and arithmetic correctly?

(b)

So for the functions f and g I just picked some arbitrary functions.

Let [tex]f(x)=e^x[/tex] and [tex]g(x)=e^{-x}[/tex]

So now,

[tex]

x^3=e^x+e^{4x}

[/tex]

And also,

[tex]

-3x^2=-e^x-e^{4x}

[/tex]

And this is where I stall. I'm not sure what to do after this point. My book doesn't have an example of solving these types of problems and the notes our professor gave us doesn't show an example either.