Math 2280:
PRACTICE TEST 2
This test is closed book, closed notes test. Only small non-graphing calculator is allowed.
1. Find a fundamental matrix and use it to find a solution satisfying the given initial data:
x′ = 3x − y,
y ′ = x + y,
x(0) = 1,
y(0) = 0
2. Transform the given system of second order equations into an equivalent system of first
order equations:
x′′ = 3x − y + 2z,
y ′′ = x + y − 4z,
z ′′ = 5x − y − z
3. Find both the steady periodic solution xsp = C cos(ωt − α) of the given differential
equation and the transient solution xtr (t) that satisfies the given initial conditions. Sketch
their plots.
x′′ + 4x′ + 5x = 10 cos(3t), x(0) = x′ (0) = 0.
4. Find general solution of the given system of first order equations and sketch typical
solution curves:
x′1 = 4x1 + x2 , x′2 = 6x1 − x2
5. Solve the initial value problem:
z ′′ + z = cos(x),
z(0) = 1,
z ′ (0) = 1
6. In the first experiment, a mass m is attached to both a spring (with given spring constant
k) and a dashpot (with given damping constant c). The mass is set in motion with initial
position x0 and initial velocity v0 . Find the position function x(t) and determine if the
motion is overdamped, critically damped or underdamped.
In the second experiment, the mass is set in motion with the same initial position and
velocity, but the dashpot is disconnected (c = 0). Find the undamped position function
u(t) = C0 cos(ω0 t − α0 ).
m = 1,
c = 8,
k = 16,
x0 = 5,
v0 = −10