Final
Be sure to show neat, organized, complete work in the space provided.
1. (10 points) Find the general solution to the differential equation
..
.
u + 4u + 4u = cos t.
2. (10 points) Using the table (not provided on practice exam) find the inverse Laplace transform L−1 (F ), where
s2 + 1
.
F (s) =
(s + 2)3
3. (10 points) A 12-lb weight is attached to a frictionless spring, which in turn is suspended from
the ceiling. The weight stretches the spring 1.5 inches and comes to rest in its equilibrium
position. The weight is then pulled down an additional 2 inches and released. Find the
resulting motion of the weight as a function of time.
4. (10 points) A 32-lb weight is attached to the lower end of a coil spring, which in turn is
suspended from the ceiling. The weight stretches the spring 2 ft in the process. The weight
is then pulled down 6 inches below its equilibrium position and released. The resistance of
the medium is given as 4 lb/(ft/sec). Determine the motion of the weight, as well as its
damped amplitude and frequency.
5. (10 points) Calculate explicitly the Laplace transform L(f ), where
f (t) = cos 2t.
6. (10 points) Use the Laplace transform to solve the initial value problem
y ′′ − 3y ′ + 2y = 0, y(0) = 1, y ′ (0) = 0.