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Midterm2 - MAE105 Second Midterm Exam(open book closed...

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MAE105 Second Midterm Exam (open book; closed notes; no computers, no calculators, no cell phones) Name:_________________________ Time: 3:35 to 4:50pm Date: May 13, 2008 PROBLEM 1: Consider the following PDE: 2 u x 2 + 2 u y 2 = 0 0< x <1 , 0< y < H , (a) (1 Point) Use the separation of variables, u ( x , y ) = φ ( x ) G ( y ), to find two ODE’s, one for φ ( x ), and the other for G ( y ), in such a way that the solution for φ ( x ) would be periodic. (b) (1 Point) Solve the two ODE’s to find general expressions for φ ( x ) and G ( y ) such that your solution for φ ( x ) is periodic. Consider the following boundary conditions: u ( x , H ) = x sin 3 π x , (*) u ( x , 0) = 0, u (0, y ) = 0, u (1 , y ) = 0. (**) (c) (0.5 Point) Apply the three boundary conditions (**) to find the corresponding boundary conditions for φ ( x ) and G ( y ). (d) (0.5 Point) Find the eigenvalues, λ n , and eigenfunctions, φ n ( x ), using the ODE for φ ( x ) and the corresponding boundary conditions. (e) (0.5 Point) Apply the boundary condition at y = 0, i.e. , G (0) = 0, to find an expression for G ( y ).
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