Exam given in term E97, July 24, 1997
Postscript 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 springmass 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 firstorder equation y' = y^{2} 
3y where are the equilibria? For each equilibrium, determine
if it is stable or if it is unstable.
Nathan Gibson
12/11/1997