Math 293 practice prelim 3

Math 293 practice prelim 3 - Math 293 Practice Prelim#3(The...

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Math 293 Practice Prelim #3 Spring 2000 (The actual exam will not be as long as this one.) Integration formulas which may be useful: Z x sin ax dx = - x cos ax a + sin ax a 2 Z x cos ax dx = x sin ax a + cos ax a 2 Z x 2 sin ax dx = - x 2 cos ax a + 2 x sin ax a 2 + 2 cos ax a 3 Z x 2 cos ax dx = x 2 sin ax a + 2 x cos ax a 2 - 2 sin ax a 3 1 . Let f ( x ) = x 2 when - 1 x 1, and let f ( x ) be periodic with period 2. (a) Graph f on the interval - 3 x 3. (b) Is f even, odd, or neither? Is f continuous? (c) For what values of x does the Fourier series of f actually converge to f ( x )? (d) Compute the Fourier series of f . 2 . Solve the following initial-boundary value problem for the heat equation. It is not necessary to give any derivation or justification, for this problem. u t = 3 u xx , u (0 , t ) = 0 , u ( π, t ) = 0 , u ( x, 0) = 3 sin x + sin 2 x - 1 4 sin 5 x. 3 . Consider the second order equation y 00 - 2 y 0 + λy = 0 with the boundary conditions y (0) = 0, y ( π ) = 0. Find all the eigenvalues λ for which the boundary value problem has nontrivial solutions, and find the nontrivial solutions. For full credit you should consider each of the three cases λ > 1, λ = 1, and λ < 1. 4 . Consider the boundary value problem, which is
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