Differential
Equations Math 285 Hour Exam 2 Fall, 2011
4. |
A mass 10 g stretches a vertically suspended spring 140 cm. The mass moves in a medium that imparts a resistance force of 40 dynes when the speed is 1 cm/sec. The mass is pulled up 4 cm and set in motion with initial downward velocity of 20 cm/sec. (All the quantities are given in the metric cm-gram-sec system of units. “Dyne” is the unit of force in this system. Recall the acceleration due to gravity is g = 980 cm/sec2. Since calculators are not to be used, the answers may include π, √, etc.) a. Set up an initial value problem for the position of the mass relative to its static equilibrium after time t. Use the downward direction of the coordinate axis. b. Solve the initial value problem in part a. c. Find the circular (pseudo) frequency, time varying amplitude, and phase of the motion. Answers: a. 10 x” + 40 x’ + 70 x = 0 x(0) = -4, x ‘ (0) = 20 b. x(t) = ( - 4 cos √3 t + 4 √3 sin √3 t ) e -2t c. ω = √3 Amp(t) = 8 e -2t α = 2π/3 |
5. |
Find a general solution to the equation x3 y” - 3 x2 y’ + 3 x y = 2 x 3 given that the functions x and x3 satisfy the corresponding homogeneous equation. (Use variation of parameters. Pay attention to the coefficient of y”.) Answer: y = -2 x2 + C1 x + C2 x 3 |