Differential Equations Final Exam Practice
Solutions
1. A tank originally contains 10 gal of water with 1/2 lb of salt in solution.
1
Water containing a salt concentration of 200 (10 − t)2 (sin(t) + 1) lb per gallon
flows into the tank at a rate of 1 gal/ min, and the mixture is allowed to
flow out of the tank at a rate of 2 gal/ min. The mixture is kept uniform
by stirring. Let Q(t) (in lb) be the amount of salt in the tank after time t
(in min).
(i) How long (in min) will it take for the tank to become empty?
Answer: t = 10 min.
(ii) Write the initial value problem for Q(t) (before the tank is empty)
and solve it.
Answer:
1
Q(t) = (10 − t)2 (t − cos(t) + 2).
200
2. Solve the differential equation
(2xy + y 3 )dx + (x2 + 3xy 2 − 2y)dy = 0.
Answer:
x2 y + xy 3 − y 2 = C.
Solutions
1. A tank originally contains 10 gal of water with 1/2 lb of salt in solution.
1
Water containing a salt concentration of 200 (10 − t)2 (sin(t) + 1) lb per gallon
flows into the tank at a rate of 1 gal/ min, and the mixture is allowed to
flow out of the tank at a rate of 2 gal/ min. The mixture is kept uniform
by stirring. Let Q(t) (in lb) be the amount of salt in the tank after time t
(in min).
(i) How long (in min) will it take for the tank to become empty?
Answer: t = 10 min.
(ii) Write the initial value problem for Q(t) (before the tank is empty)
and solve it.
Answer:
1
Q(t) = (10 − t)2 (t − cos(t) + 2).
200
2. Solve the differential equation
(2xy + y 3 )dx + (x2 + 3xy 2 − 2y)dy = 0.
Answer:
x2 y + xy 3 − y 2 = C.
