Deck 10: First-Order Differential Equations

Question 1

Match the differential equation with the appropriate slope field.

Answer 1

C

Question 2

Match the differential equation with the appropriate slope field.

Answer 2

B

Question 3

Obtain a slope field and add to its graphs of the solution curves passing through the given points.

Answer 3

Question 4

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified
increment size. Round your results to four decimal places.

A) y1 = 1.6409, y2 = 0.2704, y3 = -5.3889
B) y1 = 2.1409, y2 = 1.5704, y3 = -5.0389
C) y1 = 0.6409, y2 = -2.3296, y3 = -6.0889
D) y1 = 1.1409, y2 = -1.0296, y3 = -5.7389

Question 5

Obtain a slope field and add to its graphs of the solution curves passing through the given points.

Question 6

Match the differential equation with the appropriate slope field.

Question 7

Obtain a slope field and add to its graphs of the solution curves passing through the given points.

Question 8

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified
increment size. Round your results to four decimal places.

A) y1 = 0.4, y2 = 0.63, y3 = 0.7472
B) y1 = 0.7, y2 = 0.99, y3 = 1.9136
C) y1 = 0.5, y2 = 0.75, y3 = 1.03125
D) y1 = 0.7, y2 = 0.99, y3 = 1.2656

Question 9

Solve. Round your results to four decimal places.

Question 10

Solve. Round your results to four decimal places.

Question 11

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified
increment size. Round your results to four decimal places.
y = 2xy - 2y, y(2) = 3, dx = 0.2

A) y1 = 0.9000, y2 = 6.2160, y3 = 15.9130
B) y1 = 2.7000, y2 = 3.8850, y3 = 29.8368
C) y1 = 4.5000, y2 = 6.9930, y3 = 19.8912
D) y1 = 4.2000, y2 = 6.2160, y3 = 9.6970

Question 12

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified
increment size. Round your results to four decimal places.

A) y1 = -4.8750, y2 = -4.0800, y3 = -4.1600
B) y1 = -3.2500, y2 = -3.4000, y3 = -3.4667
C) y1 = -6.5000, y2 = -6.8000, y3 = -13.8667
D) y1 = -6.5000, y2 = -5.1000, y3 = -6.9333

Question 13

Match the differential equation with the appropriate slope field.

Question 14

Obtain a slope field and add to its graphs of the solution curves passing through the given points.

Question 15

Solve. Round your results to four decimal places.

Question 16

Match the differential equation with the appropriate slope field.

Question 17

Obtain a slope field and add to its graphs of the solution curves passing through the given points.

Question 18

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified
increment size. Round your results to four decimal places.
y = -x(1 - y), y(2) = 3, dx = 0.2

A) y1 = 3.8000, y2 = 5.0320, y3 = 6.9674
B) y1 = 7.0000, y2 = 50.3200, y3 = 69.6736
C) y1 = 0.7000, y2 = 2.5160, y3 = 3.4837
D) y1 = 2.8000, y2 = 10.0640, y3 = 13.9347

Question 19

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified
increment size. Round your results to four decimal places.

A) y1 = 3.2699, y2 = 5.1723, y3 = 8.9899
B) y1 = 4.0873, y2 = 5.9897, y3 = 9.8074
C) y1 = 3.6786, y2 = 5.5810, y3 = 9.3986
D) y1 = 4.4960, y2 = 6.3985, y3 = 10.2161

Question 20

Match the differential equation with the appropriate slope field.

Question 21

Solve the differential equation.

Question 22

Solve the differential equation.

Question 23

Solve the differential equation.

Question 24

Solve the initial value problem.

Question 25

Solve the differential equation.

Question 26

Solve the differential equation.

Question 27

Solve the problem.

Question 28

Solve the initial value problem.

Question 29

Solve the differential equation.

Question 30

Solve the problem.

Question 31

Solve the initial value problem.

Question 32

Solve the problem.

Question 33

Solve the problem.

Question 34

Solve the differential equation.

Question 35

Solve the differential equation.

Question 36

Solve the differential equation.

Question 37

Solve the problem.

Question 38

Obtain a slope field and add to its graphs of the solution curves passing through the given points.

Question 39

Solve the problem.

Question 40

Solve the differential equation.

Question 41

Solve the differential equation.

Question 42

Determine which of the following equations is correct.

Question 43

Solve the problem.
A 200 gal tank is half full of distilled water. At time = 0, a solution containing 1 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 2 gal/min. When the tank
Is full, how many pounds of concentrate will it contain?

A) 100 pounds
B) 200 pounds
C) 120 pounds
D) 150 pounds

Question 44

Solve the initial value problem.

Question 45

Solve the problem.

A) 5.01 min
B) 6.01 min
C) 8.01 min
D) 7.01 min

Question 46

Solve the problem.
A tank contains 100 gal of fresh water. A solution containing 2 lb/gal of soluble lawn fertilizer runs into the tank at the rate of 1 gal/min, and the mixture is pumped out of the tank at the rate of 2 gal/min. Find the
Maximum amount of fertilizer in the tank and the time required to reach the maximum.

A) 60 pounds, 40 minutes
B) 48 pounds, 40 minutes
C) 48 pounds, 60 minutes
D) 50 pounds, 50 minutes

Question 47

Solve the initial value problem.

Question 48

Solve the problem.
dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y
When f(t) = -8t.

Question 49

Solve the differential equation.

Question 50

Solve the initial value problem.

Question 51

Solve the initial value problem.

Question 52

Solve the problem.
A tank initially contains 100 gal of brine in which 40 lb of salt are dissolved. A brine containing 2 lb/gal of salt runs into the tank at the rate of 4 gal/min. The mixture is kept uniform by stirring and flows out of the tank at
The rate of 3 gal/min. Find the solution to the differential equation that models the mixing process.

Question 53

Solve the problem.
dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y
When f(t) = 11t.

Question 54

Solve the differential equation.

Question 55

Determine which of the following equations is correct.

Question 56

Solve the problem.
A 100 gal tank is half full of distilled water. At time = 0, a solution containing 2 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 3 gal/min. When the tank
Is full, how many pounds of concentrate will it contain?

A) 187.5 pounds
B) 200 pounds
C) 150 pounds
D) 175 pounds

Question 57

Solve the differential equation.

Question 58

Solve the initial value problem.

Question 59

Solve the problem.
A tank initially contains 120 gal of brine in which 40 lb of salt are dissolved. A brine containing 2 lb/gal of salt runs into the tank at the rate of 9 gal/min. The mixture is kept uniform by stirring and flows out of the tank at
The rate of 6 gal/min. Write, in standard form, the differential equation that models the mixing process.

Question 60

Solve the differential equation.

Question 61

Solve.

Question 62

Solve.

A) 22.26 weeks
B) 46.73 weeks
C) 26.02 weeks
D) 14.23 weeks

Question 63

Identify equilibrium values and determine which are stable and which are unstable.
y = (y - 4)(y - 6)(y - 7)

A) y = 7 is a stable equilibrium value and y = 6 and y = 4 are unstable equilibria.
B) y = 4 is a stable equilibrium value and y = 6 and y = 7 are unstable equilibria.
C) y = 6 is a stable equilibrium value and y = 4 and y = 7 are unstable equilibria.
D) y = 5, y = 4 and y = 7 are unstable equilibria.

Question 64

Show that the curves are orthogonal.

Question 65

Construct a phase line. Identify signs of y and y .

Question 66

Solve the problem.
How many seconds after the switch in an RL circuit is closed will it take the current i to reach 20% of its steady state value? Express answer in terms of R and L and round coefficient to the nearest hundredth.

A) 0.22 L/R seconds
B) 1.81 L/R seconds
C) 1.61 L/R seconds
D) 0.42 L/R seconds

Question 67

Solve.

Question 68

Solve the problem.

A) -4.60 L/R seconds
B) -4.40 L/R seconds
C) 1.25 L/R seconds
D) 1.05 L/R seconds

Question 69

Sketch several solution curves.

Question 70

Show that the curves are orthogonal.

Question 71

Construct a phase line. Identify signs of y and y .

Question 72

Identify equilibrium values and determine which are stable and which are unstable.

A) y = 4 is a stable equilibrium value and y = -4 is an unstable equilibrium.
B) y = -4 and y = 5 are stable equilibrium values.
C) y = -4 is a stable equilibrium value and y = 4 is an unstable equilibrium.
D) There are no equilibrium values.

Question 73

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.

Question 74

Identify equilibrium values and determine which are stable and which are unstable.

A) y = 4 is a stable equilibrium value and y = -5 is an unstable equilibrium.
B) y = -5 is a stable equilibrium value and y = 4 is an unstable equilibrium.
C) y = -4 is a stable equilibrium value and y = 5 is an unstable equilibrium.
D) y = 5 is a stable equilibrium value and y = 4 is an unstable equilibrium.

Question 75

Solve.
A 57-kg skateboarder on a 2-kg board starts coasting on level ground at 5 m/sec. Let k = 3.2 kg/sec. How long will it take the skater's speed to drop to 3 m/sec?

A) -9.42 sec
B) 9.10 sec
C) 0.32 sec
D) 9.42 sec

Question 76

Solve.
A 60-kg skateboarder on a 1-kg board starts coasting on level ground at 8 m/sec. Let k = 3.2 kg/sec. About how far will the skater coast before reaching a complete stop?

A) 1536.00 m
B) 152.50 m
C) 24.00 m
D) 150.00 m

Question 77

Identify equilibrium values and determine which are stable and which are unstable.

A) y = 0 is an unstable equilibrium value.
B) y = 7 is an unstable equilibrium value.
C) y = 0 is a stable equilibrium value.
D) There are no equilibrium values.

Question 78

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
y = -mx

Question 79

Identify equilibrium values and determine which are stable and which are unstable.

A) y = 0 and y = 5 are unstable equilibrium values.
B) y = 0 is a stable equilibrium value and y = 5 is an unstable equilibrium.
C) y = 5 is a stable equilibrium value and y = 0 is an unstable equilibrium.
D) There are no equilibrium values.

Question 80

Construct a phase line. Identify signs of y and y .