Math 427KPractice Final Exam12/05/2011Disclaimer: Nick wrote this practice final without any knowledge of what will be covered on the examother than it will be 8 problems and cumulative. Professor Um will write the final, so it may or maynot bear resemblance to what you see here. This is just intended as a study aid.1. Solve the differential equationyey2sin x(2 sin x)dydx+ y cos x= 0.2. Noting that yp= etis a solution to the nonhomogeneous problemy000+ y00+ y0+ y = 4et,find the general solution.3. Show that x = 0 is a regular singular point forx2y00+ xy0− 2y = 0.Then find the general solution.4. Initially, Tank A contains 2 lbs of salt dissolved in 100 gallons of water and Tank B contains 200gallons of water and no salt. The tanks are then connected in a way that allows 5 gal/s to flowfrom Tank A to Tank B and 5 gal/s to flow from Tank B to Tank A. Write down a system oflinear equations describing the flow of salt in this system with the given initial value data. Thensolve the system for the (vector of) functions giving the amount of salt in the tanks at t seconds.5. Compute a Fourier series of period 2 for the functionf(x) = x2if x ∈ [−1, 1], f(x) = f(x + 2).Use the fact that f(1) = 1 to show that∞Xn=11n2=π26.6. Solve the heat equationuxx= ut, 0 < x < 3, t > 0;u(0, t) = u(3, t) = 0, t > 0;u(x, 0) = 5 sin πx + 7 sin 2πx, 0 ≤ x ≤ 3.7. Compute the Laplace transform of eatusing the definition. Then solve the initial value problemy000− y0= u2(t), y(0) = 0, y0(0) = 0, y00(0) = 2.8. Solve the initial value problemy000+ 3y00+ 3y0+ y = 24e−t, y(0) = 0, y0(0) = 0, y00(0) =
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