Math. 2403, Practice Final
1. Solve the initial value problem
dx
t2
=
,
dt
x + t3 x
x(0) = −2.
2. A large tank is initially filled with 500 L of water containing 50 kg of salt. A brine
containing 2 kg of salt per gallon is pumped into the tank at a rate of 5 L/min. The
well-mixed solution is pumped out at the same rate. After 10 min, a leak develops in the
tank and an additional liter per minute of mixture flows out of the tank. What is the
concentration, in kilograms per liter, of salf in the tank 20 min after the leak develops?
3. Solve the initial value problem
y 00 − y = sin t − e2t ,
y(0) = 1, y 0 (0) = −1.
4. Find a general solution of

0
0

x (t) =
0
−2
5. Let
A=
1
0
−5
3
0
−2
3

0
1  x(t).
−4
.
(a) Find eAt .
(b) Find the solution of
0
x (t) = Ax(t),
2
x(0) =
.
3
(c) Draw the phase portrait for x0 (t) = Ax(t).
6. A mass-spring system contains two masses whose displacement functions x(t), y(t)
satisfy the differential equations
x00 = −4x + y,
y 00 = x − 4y.
Find the two natural frequencies of the system, its two natural modes of oscillation, and
write down a general solution of the system.
7. Find the critical points of
x0 = 2x − x2 − xy
y 0 = 3y − 2y 2 − xy
and classify their type and stability. Draw the phase portrait for the system. Indicate the
directions of the vector field at various points.
8. Find L{f (t)}, where
f (t) =
(
e−t
1
for 0 < t < 1,
for 1 < t < 2,
and f (t) has period 2.
9. Use the method of Laplace transforms to solve the initial value problem
(
y 00 + 5y 0 − 6y = 21et
y(0) = −1, y 0 (0) = 9.