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**Unformatted text preview: **u satisFes the Dirichlet boundary conditions, show that A ( t ) is a non-increasing function of t . (Hint for (a) and (c): Use an argument similar to an energy argument). 3. (a) Solve the wave equation with friction: u xx = u tt + 2 u t for 0 < x < and t > with the initial conditions u ( x, 0) = sin x , u t ( x, 0) = 0, and the boundary conditions u (0) = u ( ) = 0. (Hint: Look for separated solutions) (b) If E ( t ) = 1 2 Z u 2 t + u 2 x dx, show that lim t E ( t ) = 0 . (Hint: To do this, you can calculate E ( t ) explicitly). 4. ind as general a solution u ( x, y, z ) as you can to the third-order equation u xyz = 0...

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