384 Chapter 6: Differential Equations
6.4 First-Order Linear Differential Equations
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Solve the first order linear differential equation.
a.
b.
c.
d.
e.
____ 2. Solve the first-order linear differential equation .
a.
b.
c.
d.
e.
____ 3. Solve the first order linear differential equation.
a.
b.
c.
d.
e.
6.4 First-Order Linear Differential Equations 385
____ 4. Find the particular solution of the differential equation that
satisfies the initial condition .
a.
b.
c.
d.
e.
____ 5. Find the particular solution of the differential equation that
satisfies the boundary condition .
a.
b.
c.
d.
e.
____ 6. Find the particular solution of the differential equation that
satisfies the boundary condition .
a.
b.
c.
d.
e.
386 Chapter 6: Differential Equations
____ 7. Find the particular solution of the differential equation that
satisfies the boundary condition .
a.
b.
c.
d.
e.
____ 8. A 300-gallon tank is full of a solution containing 35 pounds of concentrate. Starting
at time distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred
solution is withdrawn at the same rate. Find the amount of concentrate Q in the solution as a function
of t.
a.
b.
c.
d.
e.
____ 9. A 200-gallon tank is full of a solution containing 25 pounds of concentrate. Starting
at time , distilled water is added to the tank at a rate of 10 gallons per minute, and the well-stirred
solution is withdrawn at the same rate. Find the time at which the amount of concentrate in the tank
reaches 15 pounds. Round your answer to one decimal place.
a. 18.3 min
b. 3.6 min
c. 20.4 min
d. 10.2 min
e. 5.1 min
6.4 First-Order Linear Differential Equations 387
____ 10. A 300-gallon tank is full of a solution containing 35 pounds of concentrate. Starting
at time , distilled water is added to the tank at a rate of 30 gallons per minute, and the well-stirred
solution is withdrawn at the same rate. Find the quantity of the concentrate in the solution as .
a. 1
b. 35
c. 36
d. 0
e. 30
____ 11. A 300-gallon tank is half full of distilled water. At time , a solution containing
0.5 pound of concentrate per gallon enters the tank at the rate of 8 gallons per minute, and the well-
stirred mixture is withdrawn at the rate of 6 gallons per minute. At what time will the tank be full?
a. 150 minutes
b. 76 minutes
c. 75 minutes
d. 301 minutes
e. 600 minutes
____ 12. A 200-gallon tank is half full of distilled water. At time , a solution containing
0.5 pound of concentrate per gallon enters the tank at the rate of 5 gallons per minute, and the well-
stirred mixture is withdrawn at the rate of 3 gallons per minute. At the time the tank is full, how many
pounds of concentrate will it contain? Round your answer to two decimal places.
a. 11.61 lbs
b. 64.64 lbs
c. 71.34 lbs
d. 49.39 lbs
e. 82.32 lbs
____ 13. Suppose an eight-pound object is dropped from a height of 5000 feet, where the air
resistance is proportional to the velocity. Write the velocity as a function of time if its velocity after 7
seconds is approximately –75 feet per second. Use a graphing utility or a computer algebra system.
Round numerical answers in your answer to four places.
a.
b.
c. 0.8018
d.
e.
388 Chapter 6: Differential Equations
____ 14. Assume an object weighing 7 pounds is dropped from a height of 9,000 feet, where
the air resistance is proportional to the velocity. Round numerical answers in your answer to two
places.
(i) Write the velocity as a function of time if the object’s velocity after 6 seconds is 3.50 feet per
second.
(ii) What is the limiting value of the velocity function?
a. (i) ; (ii) 0
b. (i) ; (ii) 0
c. (i) ; (ii) 3.5000
d. (i) ; (ii) 3.5000
e. (i) ; (ii) limit does not exist
____ 15. Solve the Bernoulli differential equation .
a.
b.
c.
d.
e.
____ 16. Find the particular solution of the differential equation passing
through the point .
a.
b.
c.
d.
e.
6.4 First-Order Linear Differential Equations 389
6.4 First-Order Linear Differential Equations
Answer Section