pp2a - Practice Prelim 2 Math 293 Fall 2007 1 Find a linear...

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Unformatted text preview: Practice Prelim 2 Math 293 Fall 2007 1. Find a linear differential operator L so that one solution of the differential equation Ly = 0 is: y(x) = 10xex + x3 ex - 2 sin(3x)ex + 10 2. The position x(t) of a mass on a spring experiencing periodic forcing obeys the equation: 2x + 4x + 52x = 2 sin(t) a) There is a transient solution of xtr (t) = 3e-2t sin(4 3t) - e-2t cos(4 3t) Express this transient solution in the form xtr (t) = Ce-t cos(1 t - ). b) Find the value of that gives practical resonance for this system. 3. a) For x > 0, use the substitution v = ln x to transform ax2 y + bxy + cy = 0 into a constant coefficient differential equation of y in terms of v. b) For x > 0, find the general solution to: x2 y + 6xy + 6y = 1 x2 4. If x2 y + 4xy + 2y = sin(x) and y () = y() = 0, find y(2). 5. Find the general solution to y (5) = y (3) - 2y - 2y . 6. Find all eigenvalues and associated eigenfunctions to the eigenvalue problem: y + y + y = 0 y(0) = y(4) = 0 7. Define f (x) on the interval (-, ) by: f (x) = Find the Fourier series of f (x). x2 0 if 0 x < if - < x 0 1 ...
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