Math 240 Final Exam Summer 2012 We have made every eort to proofread this exam for typos, etc. If you suspect that there is mistake, please state your assumption and continue to solve the problem. Or you can email your professor and he will investigate. 1. Show that the nonlinear Bernoulli's equation, dy + a(x)y = b(x)y n dx can be reduced to a linear first order equation by the substitution z = y 1 2. Find the general solution of sec n . d + = 1. d 3. Two Part: (a) Show that a linear dierential equation with variable coecients of the following type (known as Euler's or Cauchy's dierential equation): an xn dn y + an dxn 1x n 1d n 1 y dxn 1 + . . . + a1 x dy + a0 y = f (x) dx can be reduced to an equation with constant coecients by means of the substitution x = ez . (b) Solve the equation: x2 d2 y dy + 3x + 2y = log(x). dx2 dx 4. Calculate the inverse Laplace transform of: F (s) = s2 s 10s + 29) 5. Show that if L[f (t)] = F (s), then L[f (at)] = a1 F ( as ) 6. Use an appropriate infinite series method about x = 0 to find two solutions of the given dierential equation: 00 y xy 0 y = 0 7. Find a particular solution for x00 + 2 x0 + ! 2 x = A. where A is a constant force. 8. Consider the boundary value problem, 00 0 y + y = 0, y(0) = y(2), y 0 (0) = y (2) Show that except for the case when sponding to each eigenvalue. = 0, there are two independent eigenfunctions corre- 9. A mass weighing 4 pounds stretches a spring 18 inches. A periodic force equal to f (t) = cos t + sin t is impressed on the system starting at t = 0. In the absence of a damping force, for what value of will the system be in a state of pure resonance? 10. Determine the general solution of the dierential equation, 2y 000 00 + 9y + 12y 0 + 5y = 0