Exam given in term E97, July 24, 1997
Post-script version
- 1.
- Solve the initial value problems
- (a)
- x'' - 5x' + 6x = 0, x(0) = 0, x'(0) =
-1.
- (b)
-
x'' + 4x = 8 sin(2t), x(0) = 0, x'(0) =
0.
- 2.
- For the differential equation x'' + 6x' + 13x = 4 sin(2t)
- (a)
- Find the general solution to this equation.
- (b)
- What part of the solution will become unimportant after a long
time (the transient)? What part of the solution will be important
after a long time (the limiting behavior)?
- (c)
- Find the amplitude and period of the limiting solution
- 3.
- Write a differential equation that models a spring-mass system with
mass 1 kg, damping coefficient 5 kg/s, and a
spring constant 4 N/m. Is the system underdamped or
overdamped? Using the same damping coefficient and spring constant, what
mass will make the system critically damped?
- 4.
- For the first-order equation y' = y2 -
3y where are the equilibria? For each equilibrium, determine
if it is stable or if it is unstable.
Nathan Gibson
12/11/1997