Question 1

The solution of the logistic equation $\frac { d P } { d t } = P ( 8 - 2 P )$ with initial condition $P ( 0 ) = 2$ is

Accepted Answer

A)  $P = 4 / \left( 2 - e ^ { - 8 t } \right)$ 
B)  $P = 2 / \left( 8 + e ^ { - 8 t } \right)$ 
C)  $P = 8 / \left( 2 + e ^ { - 8 t } \right)$ 
D)  $P = 8 / \left( 2 - e ^ { - 8 t } \right)$ 
E)  $P = 4 / \left( 1 + e ^ { - 8 t } \right)$ 
A)  $P = 4 / \left( 2 - e ^ { - 8 t } \right)$ 
B)  $P = 2 / \left( 8 + e ^ { - 8 t } \right)$ 
C)  $P = 8 / \left( 2 + e ^ { - 8 t } \right)$ 
D)  $P = 8 / \left( 2 - e ^ { - 8 t } \right)$ 
E)  $P = 4 / \left( 1 + e ^ { - 8 t } \right)$ E

Question 2

The half-life of plutonium 239 is 24,200 years. Assume that the decay rate is proportional to the amount. An initial amount of 3 grams of radium would decay to 2 grams in approximately

Accepted Answer

A)  12200 years 
B)  14200 years 
C)  15200 years 
D)  17200 years 
E)  18200 years 
A)  12200 years 
B)  14200 years 
C)  15200 years 
D)  17200 years 
E)  18200 years B

Question 3

In the previous problem, the solution of the initial value problem is

Accepted Answer

A)  $x = 16 t ^ { 2 } + 40 t + 200$ 
B)  $x = - 16 t ^ { 2 } + 200 t + 40$ 
C)  $x = - 16 t ^ { 2 } + 40 t + 200$ 
D)  $x = 32 t ^ { 2 } + 40 t + 200$ 
E)  $x = - 32 t ^ { 2 } + 40 t + 200$ 
A)  $x = 16 t ^ { 2 } + 40 t + 200$ 
B)  $x = - 16 t ^ { 2 } + 200 t + 40$ 
C)  $x = - 16 t ^ { 2 } + 40 t + 200$ 
D)  $x = 32 t ^ { 2 } + 40 t + 200$ 
E)  $x = - 32 t ^ { 2 } + 40 t + 200$ C

Question 4

An object is taken out of a $21 ^ { \circ } \mathrm { C }$ room and placed outside where the temperature is $4 ^ { \circ } \mathrm { C }$ room. Twenty-five minutes later the temperature is $17 ^ { \circ } \mathrm { C }$ . It cools according to Newton's Law. The temperature of the object after one hour is

Accepted Answer

A)  $12.2 ^ { \circ } \mathrm { C }$ 
B)  $12.9 ^ { \circ } \mathrm { C }$ 
C)  $13.6 ^ { \circ } \mathrm { C }$ 
D)  $14.3 ^ { \circ } \mathrm { C }$ 
E)  $15.0 ^ { \circ } \mathrm { C }$ 
A)  $12.2 ^ { \circ } \mathrm { C }$ 
B)  $12.9 ^ { \circ } \mathrm { C }$ 
C)  $13.6 ^ { \circ } \mathrm { C }$ 
D)  $14.3 ^ { \circ } \mathrm { C }$ 
E)  $15.0 ^ { \circ } \mathrm { C }$

Question 5

In the previous problem, how much of X, Y, and Z are left after a long period of time?

Accepted Answer

A)  $x = 0 , y = 5 , z = 5$ 
B)  $x = 5 , y = 5 , z = 0$ 
C)  $x = 5 , y = 0 , z = 5$ 
D)  $x = 0 , y = 0 , z = 10$ 
E)  none of the above 
A)  $x = 0 , y = 5 , z = 5$ 
B)  $x = 5 , y = 5 , z = 0$ 
C)  $x = 5 , y = 0 , z = 5$ 
D)  $x = 0 , y = 0 , z = 10$ 
E)  none of the above

Question 6

A tank contains 200 gallons of water in which 300 grams of salt is dissolved. A brine solution containing 0.4 kilograms of salt per gallon of water is pumped into the tank at the rate of 5 liters per minute, and the well-stirred mixture is pumped out at the same rate. Let $A ( t )$ represent the amount of salt in the tank at time t. The correct initial value problem for $A ( t )$ is

Accepted Answer

A)  $\frac { d A } { d t } = 2 + A / 40 , A ( 0 ) = 0.3$ 
B)  $\frac { d A } { d t } = 2 - A / 40 , A ( 0 ) = 0.3$ 
C)  $\frac { d A } { d t } = 5 + A / 40 , A ( 0 ) = 300$ 
D)  $\frac { d A } { d t } = 5 - A / 40 , A ( 0 ) = 300$ 
E)  $\frac { d A } { d t } = 0.4 - A / 40 , A ( 0 ) = 300$ 
A)  $\frac { d A } { d t } = 2 + A / 40 , A ( 0 ) = 0.3$ 
B)  $\frac { d A } { d t } = 2 - A / 40 , A ( 0 ) = 0.3$ 
C)  $\frac { d A } { d t } = 5 + A / 40 , A ( 0 ) = 300$ 
D)  $\frac { d A } { d t } = 5 - A / 40 , A ( 0 ) = 300$ 
E)  $\frac { d A } { d t } = 0.4 - A / 40 , A ( 0 ) = 300$

Question 7

In the Lotka-Volterra predator-prey model $\frac { d x } { d t } = - a x + b x y , \frac { d y } { d t } = e y - c x y$ , where $x ( t )$ is the predator population and $y ( t )$ is the prey population, the coefficient c represents which of the following:

Accepted Answer

A)  the predator die-off rate 
B)  the prey growth rate 
C)  the increase in the predator population due to interactions with the prey 
D)  the decrease in the prey population due to interactions with the predator 
E)  none of the above 
A)  the predator die-off rate 
B)  the prey growth rate 
C)  the increase in the predator population due to interactions with the prey 
D)  the decrease in the prey population due to interactions with the predator 
E)  none of the above

Question 8

The population of a certain town doubles in 14 years. How long will it take for the population to triple? Assume that the rate of increase of the population is proportional to the population.

Accepted Answer

A)  18.2 years 
B)  20.2 years 
C)  22.2 years 
D)  23.2 years 
E)  24.2 years 
A)  18.2 years 
B)  20.2 years 
C)  22.2 years 
D)  23.2 years 
E)  24.2 years

Question 9

In the logistic model for population growth, $\frac { d P } { d t } = P ( 12 - 3 P )$ , what is the carrying capacity of the population $P ( t )$ ?

Accepted Answer

A)  4 
B)  $1 / 4$ 
C)  12 
D)  3 
A)  4 
B)  $1 / 4$ 
C)  12 
D)  3

Question 10

In the logistic model for population growth, $\frac { d P } { d t } = P ( 8 - 2 P )$ , the carrying capacity of the population $P ( t )$ is

Accepted Answer

A)  8 
B)  2 
C)  4 
D)  $1 / 4$ 
E)  16 
A)  8 
B)  2 
C)  4 
D)  $1 / 4$ 
E)  16

Question 11

Tank A contains 80 gallons of water in which 20 pounds of salt has been dissolved. Tank B contains 30 gallons of water in which 5 pounds of salt has been dissolved. A brine mixture with a concentration of 0.5 pounds of salt per gallon of water is pumped into tank A at the rate of 4 gallons per minute. The well-mixed solution is then pumped from tank A to tank B at the rate of 6 gallons per minute. The solution from tank B is also pumped through another pipe into tank A at the rate of 2 gallons per minute, and the solution from tank B is also pumped out of the system at the rate of 4 gallons per minute. The correct differential equations with initial conditions for the amounts, $x ( t )$ and $y ( t )$ , of salt in tanks A and B, respectively, at time t are

Accepted Answer

A)  $\frac { d x } { d t } = 2 - x / 40 + y / 5 , \frac { d y } { d t } = x / 40 - y / 3 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
B)  $\frac { d x } { d t } = 2 - 3 x / 40 + y / 15 , \frac { d y } { d t } = 3 x / 40 - y / 5 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
C)  $\frac { d x } { d t } = 4 - 3 x / 40 + y / 15 , \frac { d y } { d t } = 3 x / 40 - y / 5 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
D)  $\frac { d x } { d t } = 4 - x / 40 + y / 5 , \frac { d y } { d t } = x / 40 - y / 3 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
E)  $\frac { d x } { d t } = 2 - 3 x / 40 + y / 5 , \frac { d y } { d t } = x / 40 - y / 5 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
A)  $\frac { d x } { d t } = 2 - x / 40 + y / 5 , \frac { d y } { d t } = x / 40 - y / 3 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
B)  $\frac { d x } { d t } = 2 - 3 x / 40 + y / 15 , \frac { d y } { d t } = 3 x / 40 - y / 5 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
C)  $\frac { d x } { d t } = 4 - 3 x / 40 + y / 15 , \frac { d y } { d t } = 3 x / 40 - y / 5 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
D)  $\frac { d x } { d t } = 4 - x / 40 + y / 5 , \frac { d y } { d t } = x / 40 - y / 3 , x ( 0 ) = 20 , y ( 0 ) = 5$ 
E)  $\frac { d x } { d t } = 2 - 3 x / 40 + y / 5 , \frac { d y } { d t } = x / 40 - y / 5 , x ( 0 ) = 20 , y ( 0 ) = 5$

Question 12

In the previous problem, the amount of chemical $C , X ( t )$ , produced by time t is

Accepted Answer

A)  $x = 800 \left( 1 - e ^ { - 400 k t } \right) / \left( 2 - e ^ { - 400 k t } \right) , \text { where } k = \ln ( 29 / 26 ) / 4000$ 
B)  $x = 800 \left( 1 - e ^ { - 800 k t } \right) / \left( 4 - e ^ { - 800 k t } \right) , \text { where } k = \ln ( 29 / 20 ) / 8000$ 
C)  $x = 400 \left( 1 - e ^ { - 800 k t } \right) / \left( 4 - e ^ { - 800 k t } \right) , \text { where } k = \ln ( 29 / 20 ) / 8000$ 
D)  $x = 400 \left( 1 - e ^ { - 400 k t } \right) / \left( 4 - e ^ { - 400 k t } \right) , \text { where } k = \ln ( 29 / 26 ) / 4000$ 
E)  $x = 600 \left( 1 - e ^ { - 800 k t } \right) / \left( 4 - e ^ { - 800 k t } \right) , \text { where } k = \ln ( 29 / 20 ) / 8000$ 
A)  $x = 800 \left( 1 - e ^ { - 400 k t } \right) / \left( 2 - e ^ { - 400 k t } \right) , \text { where } k = \ln ( 29 / 26 ) / 4000$ 
B)  $x = 800 \left( 1 - e ^ { - 800 k t } \right) / \left( 4 - e ^ { - 800 k t } \right) , \text { where } k = \ln ( 29 / 20 ) / 8000$ 
C)  $x = 400 \left( 1 - e ^ { - 800 k t } \right) / \left( 4 - e ^ { - 800 k t } \right) , \text { where } k = \ln ( 29 / 20 ) / 8000$ 
D)  $x = 400 \left( 1 - e ^ { - 400 k t } \right) / \left( 4 - e ^ { - 400 k t } \right) , \text { where } k = \ln ( 29 / 26 ) / 4000$ 
E)  $x = 600 \left( 1 - e ^ { - 800 k t } \right) / \left( 4 - e ^ { - 800 k t } \right) , \text { where } k = \ln ( 29 / 20 ) / 8000$

Question 13

The solution of the equation $\frac { d P } { d t } = P ( 12 - 3 P )$ with initial condition $P ( 0 ) = 3$ is

Accepted Answer

A)  $P = 12 / \left( 3 + e ^ { - 12 t } \right)$ 
B)  $P = 4 / \left( 3 + e ^ { - 12 t } \right)$ 
C)  $P = 4 / \left( 3 - e ^ { - 12 t } \right)$ 
D)  $P = 3 / \left( 12 + e ^ { - 12 t } \right)$ 
E)  $P = 3 / \left( 4 + e ^ { - 12 t } \right)$ 
A)  $P = 12 / \left( 3 + e ^ { - 12 t } \right)$ 
B)  $P = 4 / \left( 3 + e ^ { - 12 t } \right)$ 
C)  $P = 4 / \left( 3 - e ^ { - 12 t } \right)$ 
D)  $P = 3 / \left( 12 + e ^ { - 12 t } \right)$ 
E)  $P = 3 / \left( 4 + e ^ { - 12 t } \right)$

Question 14

Radioactive element X decays to element Y with decay constant $- 0.5$ . Y, in turn, decays to stable element Z with decay constant $- 0.1$ . What is the system of differential equations for the amounts, $x ( t ) , y ( t ) , z ( t )$ of the elements X, Y, Z, respectively, at time t, if the initial conditions are $x ( 0 ) = 10 , y ( 0 ) = 0 , z ( 0 ) = 0$ .

Accepted Answer

A)  $\frac { d x } { d t } = - 0.5 x , \frac { d y } { d t } = 0.1 x - 0.5 y , \frac { d z } { d t } = 0.2 y$ 
B)  $\frac { d x } { d t } = - 0.5 x , \frac { d y } { d t } = 0.5 x - 0.1 y , \frac { d z } { d t } = 0.5 y$ 
C)  $\frac { d x } { d t } = - 0.5 x , \frac { d y } { d t } = 0.5 x - 0.1 y , \frac { d z } { d t } = 0.1 y$ 
D)  $\frac { d x } { d t } = - 0.1 x , \frac { d y } { d t } = 0.1 x - 0.5 y , \frac { d z } { d t } = 0.1 y$ 
E)  $\frac { d x } { d t } = - 0.1 y , \frac { d y } { d t } = 0.5 x - 0.1 z , \frac { d z } { d t } = 0.5 y$ 
A)  $\frac { d x } { d t } = - 0.5 x , \frac { d y } { d t } = 0.1 x - 0.5 y , \frac { d z } { d t } = 0.2 y$ 
B)  $\frac { d x } { d t } = - 0.5 x , \frac { d y } { d t } = 0.5 x - 0.1 y , \frac { d z } { d t } = 0.5 y$ 
C)  $\frac { d x } { d t } = - 0.5 x , \frac { d y } { d t } = 0.5 x - 0.1 y , \frac { d z } { d t } = 0.1 y$ 
D)  $\frac { d x } { d t } = - 0.1 x , \frac { d y } { d t } = 0.1 x - 0.5 y , \frac { d z } { d t } = 0.1 y$ 
E)  $\frac { d x } { d t } = - 0.1 y , \frac { d y } { d t } = 0.5 x - 0.1 z , \frac { d z } { d t } = 0.5 y$

Question 15

In the previous problem, the amount of chemical $C , X ( t )$ , produced by time t is

Accepted Answer

A)  $x = 2000 \left( 1 - e ^ { - 25 k t / 3 } \right) / \left( 16 - 15 e ^ { - 25 k t / 3 } \right) , \text { where } k = 3 \ln ( 5 / 4 ) / 250$ 
B)  $x = 2000 \left( 1 - e ^ { - 125 k t } \right) / \left( 4 - e ^ { - 125 k t } \right) , \text { where } k = \ln ( 19 / 16 ) / 1250$ 
C)  $x = 400 \left( 1 - e ^ { - 25 k t / 3 } \right) / \left( 3 - e ^ { - 25 k t / 3 } \right) , \text { where } k = 3 \ln ( 3 ) / 250$ 
D)  $x = 400 \left( 1 - e ^ { - 125 k t } \right) / \left( 3 - e ^ { - 125 k t } \right) , \text { where } k = \ln ( 3 ) / 1250$ 
E)  $x = 800 \left( 1 - e ^ { - 25 k t / 3 } \right) / \left( 4 - e ^ { - 25 k t / 3 } \right) , \text { where } k = 3 \ln ( 7 / 4 ) / 250$ 
A)  $x = 2000 \left( 1 - e ^ { - 25 k t / 3 } \right) / \left( 16 - 15 e ^ { - 25 k t / 3 } \right) , \text { where } k = 3 \ln ( 5 / 4 ) / 250$ 
B)  $x = 2000 \left( 1 - e ^ { - 125 k t } \right) / \left( 4 - e ^ { - 125 k t } \right) , \text { where } k = \ln ( 19 / 16 ) / 1250$ 
C)  $x = 400 \left( 1 - e ^ { - 25 k t / 3 } \right) / \left( 3 - e ^ { - 25 k t / 3 } \right) , \text { where } k = 3 \ln ( 3 ) / 250$ 
D)  $x = 400 \left( 1 - e ^ { - 125 k t } \right) / \left( 3 - e ^ { - 125 k t } \right) , \text { where } k = \ln ( 3 ) / 1250$ 
E)  $x = 800 \left( 1 - e ^ { - 25 k t / 3 } \right) / \left( 4 - e ^ { - 25 k t / 3 } \right) , \text { where } k = 3 \ln ( 7 / 4 ) / 250$

Question 16

In the competition model $\frac { d x } { d t } = a x - b x y , \frac { d y } { d t } = c y - d x y$ where $x ( t )$ and $y ( t )$ are the populations of the competing species, moose and deer, respectively, the coefficient d represents which of the following:

Accepted Answer

A)  the moose growth rate 
B)  the deer growth rate 
C)  the decrease in the moose population due to interactions with the deer 
D)  the decrease in the deer population due to interactions with the moose 
E)  none of the above 
A)  the moose growth rate 
B)  the deer growth rate 
C)  the decrease in the moose population due to interactions with the deer 
D)  the decrease in the deer population due to interactions with the moose 
E)  none of the above

Question 17

Two chemicals, A and B, are combined, forming chemical C. The rate of the reaction is jointly proportional to the amounts of A and B not yet converted to C. Initially, there are 50 grams of A and 80 grams of B, and, during the reaction, for each two grams of A used up in the conversion, there are three grams of B used up. An experiments shows that 100 grams of C are produced in the first ten minutes. After a long period of time, how much of A and of B remains, and how much of C has been produced?

Accepted Answer

A)  30 grams of A, 0 grams of B, 100 grams of C 
B)  0 grams of A, 30 grams of B, 100 grams of C 
C)  10 grams of A, 0 grams of B, 120 grams of C 
D)  0 grams of A, 5 grams of B, 125 grams of C 
E)  0 grams of A, 0 grams of B, 130 grams of C 
A)  30 grams of A, 0 grams of B, 100 grams of C 
B)  0 grams of A, 30 grams of B, 100 grams of C 
C)  10 grams of A, 0 grams of B, 120 grams of C 
D)  0 grams of A, 5 grams of B, 125 grams of C 
E)  0 grams of A, 0 grams of B, 130 grams of C

Question 18

In the previous problem, how much salt will there be in the tank after a long period of time?

Accepted Answer

A)  2 pounds 
B)  50 pounds 
C)  75 pounds 
D)  200 pounds 
E)  none of the above 
A)  2 pounds 
B)  50 pounds 
C)  75 pounds 
D)  200 pounds 
E)  none of the above

Question 19

A bacteria culture doubles in size in 8 hours. How long will it take for the size to triple? Assume that the rate of increase of the culture is proportional to the size.

Accepted Answer

A)  12.7 hours 
B)  13.1 hours 
C)  13.5 hours 
D)  13.9 hours 
E)  14.3 hours 
A)  12.7 hours 
B)  13.1 hours 
C)  13.5 hours 
D)  13.9 hours 
E)  14.3 hours

Question 20

In the Lotka-Volterra predator-prey model $\frac { d x } { d t } = - a x + b x y , \frac { d y } { d t } = e y - c x y$ , where $x ( t )$ is the predator population and $y ( t )$ is the prey population, the coefficient e represents which of the following:

Accepted Answer

A)  the predator die-off rate 
B)  the prey growth rate 
C)  the increase in the predator population due to interactions with the prey 
D)  the decrease in the prey population due to interactions with the predator 
E)  none of the above 
A)  the predator die-off rate 
B)  the prey growth rate 
C)  the increase in the predator population due to interactions with the prey 
D)  the decrease in the prey population due to interactions with the predator 
E)  none of the above

The solution of the logistic equation $\frac { d P } { d t } = P ( 8 - 2 P )$ with initial condition $P ( 0 ) = 2$ is

The half-life of plutonium 239 is 24,200 years. Assume that the decay rate is proportional to the amount. An initial amount of 3 grams of radium would decay to 2 grams in approximately

In the previous problem, the solution of the initial value problem is

In the previous problem, how much of X, Y, and Z are left after a long period of time?

In the Lotka-Volterra predator-prey model $\frac { d x } { d t } = - a x + b x y , \frac { d y } { d t } = e y - c x y$ , where $x ( t )$ is the predator population and $y ( t )$ is the prey population, the coefficient c represents which of the following:

The population of a certain town doubles in 14 years. How long will it take for the population to triple? Assume that the rate of increase of the population is proportional to the population.

In the logistic model for population growth, $\frac { d P } { d t } = P ( 12 - 3 P )$ , what is the carrying capacity of the population $P ( t )$ ?

In the logistic model for population growth, $\frac { d P } { d t } = P ( 8 - 2 P )$ , the carrying capacity of the population $P ( t )$ is

In the previous problem, the amount of chemical $C , X ( t )$ , produced by time t is

The solution of the equation $\frac { d P } { d t } = P ( 12 - 3 P )$ with initial condition $P ( 0 ) = 3$ is

In the previous problem, the amount of chemical $C , X ( t )$ , produced by time t is

In the competition model $\frac { d x } { d t } = a x - b x y , \frac { d y } { d t } = c y - d x y$ where $x ( t )$ and $y ( t )$ are the populations of the competing species, moose and deer, respectively, the coefficient d represents which of the following:

In the previous problem, how much salt will there be in the tank after a long period of time?

A bacteria culture doubles in size in 8 hours. How long will it take for the size to triple? Assume that the rate of increase of the culture is proportional to the size.

In the Lotka-Volterra predator-prey model $\frac { d x } { d t } = - a x + b x y , \frac { d y } { d t } = e y - c x y$ , where $x ( t )$ is the predator population and $y ( t )$ is the prey population, the coefficient e represents which of the following:

Introduction to Differential Equations

First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

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Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

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Exam 3: Modeling With First-Order Differential Equations

The solution of the logistic equation dPdt=P(8−2P)\frac { d P } { d t } = P ( 8 - 2 P )dtdP​=P(8−2P) with initial condition P(0)=2P ( 0 ) = 2P(0)=2 is

The half-life of plutonium 239 is 24,200 years. Assume that the decay rate is proportional to the amount. An initial amount of 3 grams of radium would decay to 2 grams in approximately

In the previous problem, the solution of the initial value problem is

In the previous problem, how much of X, Y, and Z are left after a long period of time?

The population of a certain town doubles in 14 years. How long will it take for the population to triple? Assume that the rate of increase of the population is proportional to the population.

In the logistic model for population growth, dPdt=P(12−3P)\frac { d P } { d t } = P ( 12 - 3 P )dtdP​=P(12−3P) , what is the carrying capacity of the population P(t)P ( t )P(t) ?

In the logistic model for population growth, dPdt=P(8−2P)\frac { d P } { d t } = P ( 8 - 2 P )dtdP​=P(8−2P) , the carrying capacity of the population P(t)P ( t )P(t) is

In the previous problem, the amount of chemical C,X(t)C , X ( t )C,X(t) , produced by time t is

The solution of the equation dPdt=P(12−3P)\frac { d P } { d t } = P ( 12 - 3 P )dtdP​=P(12−3P) with initial condition P(0)=3P ( 0 ) = 3P(0)=3 is

In the previous problem, the amount of chemical C,X(t)C , X ( t )C,X(t) , produced by time t is

In the previous problem, how much salt will there be in the tank after a long period of time?

A bacteria culture doubles in size in 8 hours. How long will it take for the size to triple? Assume that the rate of increase of the culture is proportional to the size.

Introduction to Differential Equations

First-Order Differential Equations

Higher-Order Differential Equations

Modeling With Higher-Order Differential Equations

Series Solutions of Linear Equations

Laplace Transform

Systems of Linear First-Order Differential Equations

Numerical Solutions of Ordinary Differential Equations

Plane Autonomous Systems

Orthogonal Functions and Fourier Series

Boundary-Value Problems in Rectangular Coordinates

Boundary-Value Problems in Other Coordinate Systems

Integral Transform Method

Numerical Solutions of Partial Differential Equations

Mathematics Problems: Differential Equations and Linear Algebra

Mathematical Problems and Solutions

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The solution of the logistic equation $\frac { d P } { d t } = P ( 8 - 2 P )$ with initial condition $P ( 0 ) = 2$ is

In the logistic model for population growth, $\frac { d P } { d t } = P ( 12 - 3 P )$ , what is the carrying capacity of the population $P ( t )$ ?

In the logistic model for population growth, $\frac { d P } { d t } = P ( 8 - 2 P )$ , the carrying capacity of the population $P ( t )$ is

In the previous problem, the amount of chemical $C , X ( t )$ , produced by time t is

The solution of the equation $\frac { d P } { d t } = P ( 12 - 3 P )$ with initial condition $P ( 0 ) = 3$ is

In the previous problem, the amount of chemical $C , X ( t )$ , produced by time t is