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Another PDE question Where do I begin?

  1. Jul 25, 2008 #1
    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

    [tex]\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[/tex]

    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

    [tex]\frac{\partial^{2}u }{\partial x^{2}}=LC\frac{\partial^{2} u }{\partial t^{2}}+(RC+LG)\frac{\partial u }{\partial t}+RGu [/tex]

    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?

  2. jcsd
  3. Jul 26, 2008 #2
    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.
  4. Jul 26, 2008 #3
    Is it always safe to assume that

    [tex]\frac{\partial^{2}I}{\partial x \partial t}=\frac{\partial^{2}I}{\partial t \partial x}[/tex]
  5. Jul 26, 2008 #4
    It is okay as long as the mixed derivatives are continuous.
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