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Partial Differential Equations - Characteristics and Solutions

## Question

1.(a) Find characteristics for the PDE describing u(t, x) for t ? R and x ? R

Hence find the general solution.
(b) Find the solution that satisfies the initial conditions

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2. Consider the following equation for u(x,y)

Ã¯Â¿Â½ (a) Assuming x 6=0classify this equation and show that its characteristic equations are given by

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(b) Use these characteristics to show that the equation reduces to the standard form:

Ã¯Â¿Â½ Ã¯Â¿Â½(c) Hence find the general solution for u(x,y).

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3. solution to the Telegraph equation,

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can be obtained by letting E(x,t) = v(t)U (x,t)with a view to removing the first order time derivative.Use this process to obtain the Klein-Gordon equation,

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