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220-ap

# 220-ap - MATH 220 PRACTICE MIDTERM This is a closed book...

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MATH 220: PRACTICE MIDTERM This is a closed book, closed notes, no calculators exam. There are 5 problems. Solve all of them. Total score: 100 points. Problem 1. (i) (13 points) Find the general C 1 solution of the PDE 3 y 2 u x + u y = 0 . (ii) (6 points) Solve the initial value problem with initial condition u ( x, 0) = x 2 . Problem 2. (i) (10 points) Find the general C 2 solution of the PDE u xx + 3 u xt + 2 u tt = 0 . (ii) (10 points) Solve the initial value problem with initial condition u ( x, - x ) = φ ( x ) , u t ( x, - x ) = ψ ( x ) , with φ,ψ given. Problem 3. (20 points) Find the bounded solution of Laplace’s equation on the infinite strip R x × (0 , 1) y : u xx + u yy = 0 , x R , y (0 , 1) , u ( x, 0) = φ ( x ) , x R , u ( x, 1) = 0 , x R , where φ ( x ) = xe - x for x 0, φ ( x ) = 0 for x< 0. You may leave your solution as the inverse Fourier transform of a function that you have evaluated explicitly. Problem 4. (17 points) Consider the damped wave equation on R x × [0 , ) t : u tt + a ( x ) u t = ( c ( x ) 2 u x ) x , where a 0, c> 0 depending on x only, and there are constants c 1 ,c 2 > 0 such that c 1 c ( x ) c 2 for all x . Suppose that u ( x, 0) and u t ( x, 0) vanish for | x | ≥ R . Let E ( t ) = 1 2 integraldisplay R ( u t ( x,t ) 2 + c ( x ) 2 u x ( x,t ) 2 ) dx. Show that E is a decreasing (i.e. non-increasing) function of

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220-ap - MATH 220 PRACTICE MIDTERM This is a closed book...

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