Exam given in term E97, July 24, 1997

Post-script version

Solve the initial value problems
x'' - 5x' + 6x = 0, x(0) = 0,   x'(0) = -1.
x'' + 4x = 8 sin(2t), x(0) = 0,   x'(0) = 0.

For the differential equation x'' + 6x' + 13x = 4 sin(2t)
Find the general solution to this equation.
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)?
Find the amplitude and period of the limiting solution

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?

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