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Unformatted text preview: M ATH 512-M IDTERM 2 February 24, 2012, 9:30 AM - 10:18 AM No books, notes or calculators. Two pages of formulas are provided. Answer all questions. Show work for all problems. Simplify your answers. NAME: P ROBLEM Y OUR S CORE O UT OF 1 20 2 20 3 20 T OTAL 60 1. (20 points) Consider the following initial-boundary-value problem: u tt = u xx for < x < π, t > u x (0 ,t ) = 0 , u x ( π,t ) = 0 , for t > u ( x, 0) = x, u t ( x, 0) = 0 , for < x < π. By separating variables, derive a two-point boundary-value problem for an ordinary differential equation, and determine the unique solution. Solution (similar to example 1, Section 3.6): We look for solutions of the form u ( x,t ) = X ( x ) T ( t ) . We substitute into the equation to get T ′′ ( t ) X ( x ) = X ′′ ( x ) T ( t ) , and separate the variables T ′′ ( t ) T ( t ) = X ′′ ( x ) X ( x ) = k. We therefore have the fol- lowing ODEs: X ′′ ( x ) − kX ( x ) = 0 , T ′′ ( t ) − kT ( t ) = 0 . Before we solve them, we need boundary conditions. We use u ( x,t ) = X ( x ) T ( t ) and substitute it into the boundary conditions of the PDE. We get X ′ (0) T ( t ) = 0 = X ′ ( π ) T ( t ) , if T ( t ) = 0 we have the trivial solution u ( x,t ) = 0 which is not acceptable, so X ′ (0) = X ′ ( π ) = 0 ....
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