Another PDE question Where do I begin?

[physmurf]

Homework Statement

Consider an electrical cable running along the x-axis which is not well insulated from ground, so that leakage occurs along its entire length. Let V(x,t) and I(x,t) denote the voltage and current at point x in the wire at time t. These functions are related to each other by the system

$$\frac{\partial V}{\partial x}=-L \ \frac{\partial I}{\partial t}- RI, \ and \ \frac{\partial I}{\partial x}=-C\\\frac{\partial V}{\partial t}-GV$$

where L is the inductance, R is the resistance, C is the capacitance, and G is the leakage to ground. Show that V and I each satisfy

$$\frac{\partial^{2}u }{\partial x^{2}}=LC\frac{\partial^{2} u }{\partial t^{2}}+(RC+LG)\frac{\partial u }{\partial t}+RGu$$

which is called the telegraph equation.

Homework Equations

The Attempt at a Solution

I am not sure of even where to start. Initially I would think a change in variable might do it, but then again, I don't know what change in variable to use. Any suggested course of action?

Thanks.

 

[badphysicist]

Try taking partial derivatives of the equations above.. for instance try taking the partial derivative with respect to x of the first equation and then look at the other equation and see what you might want to take a partial derivative of that so that you can combine the equations.

 

[physmurf]

Is it always safe to assume that

$$\frac{\partial^{2}I}{\partial x \partial t}=\frac{\partial^{2}I}{\partial t \partial x}$$

 

[badphysicist]

It is okay as long as the mixed derivatives are continuous.