# Another PDE question Where do I begin?

1. Jul 25, 2008

### physmurf

1. The problem statement, all variables and given/known data
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.

2. Relevant equations

3. 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.

2. Jul 26, 2008

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.

3. Jul 26, 2008

### physmurf

Is it always safe to assume that

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

4. Jul 26, 2008