Differential Equations Exam #2 Spring 2004 Name_________________________________ Show all your work neatly and in numerical order on notebook paper. DO NOT WRITE ON THE BACKS OF YOUR PAGES. c 2 e 3 x is the complementary function for y y 6 y g ( x) , write the form of the particular solution for the given values of g (x ) . Do not solve. 1. Given that y c c1e a. g ( x) sin 4 x b. g ( x) xe3 x 2 x 2. Solve: y 4 y' ' ' y' ' 0 3. Solve: 5 y y 6 x 4. a. Find a linear differential operator that annihilates 13 x 9 x sin 4 x . 2 3 5x b. Find a linear differential operator that annihilates x e 5. Use variation of parameters to solve: 6. Solve: . y 3 y 2 y sin e x d2y dy 4x 8x y 0 2 dx dx 2 7. Solve the system: Dx D 2 y 0 D 3x 2 y 0 8. Solve using the substitution u y : y 1y y 2 9. A 12-pound weight stretches a spring 2 feet. The weight is released from a point 1 foot below the equilibrium position with an upward velocity of 4 ft/sec. a. Find the equation describing the resulting simple harmonic motion. b. At what time does the weight return to the point 1 foot below equilibrium position? 10. Find the eigenvalues and eigenfunctions for y y 0, y (0) 0, y 0 . 4