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Homework O: Due Monday September 11th Problem 1. Solve the following differential equations: (a) (b) dy dx dy dx + x2 y = x2 with y(0) = 1. = x2 ey with y(1) = 1. Does the solution approach infinity at a finite value of x? Problem 2. Find the explicit solution to the boundary value problem x2 y 00 − xy 0 + 4y = 0 , y(1) = 0 , y(2) = 1 . Problem 3. Calculate the transient solution and the steady-state solution for y 00 + y 0 + y = sin(ωt) , y(0) = 1 , y 0 (0) = 0 , where > 0 is constant. Write the steady-state solution in the form y = A cos(ωt − φ) for some A and φ depending on ω. Plot A versus ω for a fixed small. Problem 4. Calculate the eigenvalues and eigenfunctions for the following Sturm-Liouville problem for y(x): y 00 + λy = 0 , 0 < x < L, with the either of the three sets of boundary conditions: (a) Absorbing Boundary Conditions: y(0) = y(L) = 0, (b) No-flux Boundary Conditions: y 0 (0) = y 0 (L) = 0, (c) Periodic Boundary Conditions: y(0) = y(L) and y 0 (0) = y 0 (L) . Problem 5. Calculate the solution for x = x(t) and y = y(t) to dy = −y , dt dx = y + y 2x , dt with y(0) = 1 and x(0) = 2. In addition, determine y as a function of x by eliminating t.