Deck 10: Introduction to Differential Equations

Question 1

Find all the curves such that the tangent line at any point on the curve has a y-intercept equal to .

Answer 1

Question 2

Solve the differential equation.

Answer 2

Question 3

Solve the differential equation.

Answer 3

Question 4

Consider the circuit shown in the following figure. At the switch is closed and the current passes through the circuit.
The constant voltage is the sum of the voltage across the resistor and the voltage across the inductor.

A) Set up an initial value problem satisfied by .
B) Solve for the current .
C) What is the current after a long time?

Question 5

Solve the initial value problems.

A)
B)

Question 6

Solve the initial value problem.

Question 7

A certain chemical dissolves in water at a rate proportional to the product of the amount of chemical that had not yet been dissolved and the difference between its concentration in a saturated solution and its current concentration.
It is known that 100 g of the chemical are dissolved in 200 g of saturated solution. If 60 g of the chemical are added to 200 g of water, 20 g are dissolved in 2 h.
How many grams of the chemical are dissolved in 4 h?

Question 8

An object of constant mass is projected away from the earth with initial velocity in a direction perpendicular to the earth's surface. Assuming there is no air resistance and considering the variation of the earth's gravitational field as a function of the altitude above the earth's surface, the velocity of the object satisfies the differential equation , where is the radius of the earth.

A) Solve for the velocity as a function of .
B) What is the maximum altitude of the object if its initial velocity is ?
C) Find the escape velocity, that is, the least initial velocity for which the body will not return to the earth.
(Hint: find such that .)

Question 9

Solve the differential equations using separation of variables.

A)
B)

Question 10

Find all the curves with the following property:
the y-intercept of the tangent line at any point on the curve is equal to .

Question 11

A conical tank filled with water has a height of 7 m and a top radius of 2 m. Water leaks through a hole of area at the bottom.

A) Find the water level at time .
B) How long does it takes for the tank to empty?

Question 12

Solve the differential equations.

A)
B) . (Hint: factor the denominator.)

Question 13

The horizontal cross sections at height of a tank are discs of radius . The height of the tank is 10 m.
The tank is filled with water, and the water drains through a square hole with a side of 10 cm at the bottom of the tank.
How long does it take for the tank to empty?

Question 14

Solve the initial value problem.

Question 15

A water tank is obtained by rotating the graph of for about the y-axis (assume length is measured in meters).
The tank is filled with water and water drains through a circular hole of radius
1 cm at the bottom of the tank.
How long does it take for the tank to empty?

Question 16

Solve the initial value problems.

A)
B)

Question 17

Find all curves such that the tangent line at any point on the curve has a y-intercept that is equal to .

Question 18

Solve the initial value problem.

Question 19

Solve the initial value problems.

A)
B)

Question 20

A tank in the shape of a prism and filled with water has a height of 5 m. The cross sections are equilateral triangles with a side of 1 m. Water leaks through a hole of area 0.01 at the bottom of the tank.

A) Find the water level at time .
B) How long does it take for the tank to empty?

Question 21

The graph of an increasing function passes through the origin, and the arc length between the points and on the graph is .

A) Write an initial value problem satisfied by .
B) Use Euler's method with to estimate .

Question 22

Use Euler's method with to approximate , where is the solution of the initial value problem . Give your answer to five decimal places.

Question 23

Let be the population of a certain animal species that satisfies the logistic equation .

A) Find the equilibrium solutions and classify them.
B) What is the long-term behavior of the population?
C) Find the value of at the inflection point and explain its meaning referring to the population growth.
D) Solve the logistic equation and verify your answer to B.

Question 24

Consider the logistic equation . The solution with the following initial condition is decreasing to minus infinity at a finite value of :

A)
B)
C)
D)
E) none of the above.

Question 25

Let be an increasing function passing through such that the length of the tangent line between the tangency point and the y-axis is .

A) Find an initial value problem satisfied by .
B) Use Euler's method with to approximate .

Question 26

Use Euler's method with to approximate where is the solution of . Give your answer to five decimal places.

Question 27

Match the direction fields with their differential equations

A)
B)
C)
D)
i)
ii)
iii)
iv)

Question 28

Consider the initial value problem .

A) Use Euler's method with to approximate . Give your answer to six decimal places.
B) Solve the initial value problem and find to six decimal places.
C) Compute the error in approximating .

Question 29

Use Euler's method with step size to approximate where is the solution to the initial value problem .

Question 30

Use Euler's method with step size to approximate where is the solution to the initial value problem .

Question 31

Consider the initial value problem .

A) Use Euler's Method with to approximate
B) Use Euler's Method with to approximate
C) Solve the initial value problem and estimate the errors in parts A and B.

Question 32

Match the differential equation with its slope field.

A)
B)
C)
D)
E)
i)
ii)
iii)
iv)
v)

Question 33

Consider the initial value problem .

A) Use Euler's method with to approximate the solution at the point .
B) Solve the initial value problem and find the exact solution.

Question 34

Use Euler's method with to approximate , where is the solution of the initial value problem .

Question 35

Which of the following graphs depicts possible solutions of a logistic equation?

A)
B)
C)
D)
E) none of the above.

Question 36

Use Euler's method with step size to approximate where is the solution to the initial value problem .

Question 37

Match the differential equation with its slope field.

A)
B)
C)
i) ii)
iii)

Question 38

Consider the initial value problem

A) Use Euler's method with to approximate . Give your answer to five decimal places.
B) Solve the initial value problem and compute to five decimal places.
C) Compute the error in the approximation of .

Question 39

Consider the initial value problem. .
A) Use Euler's method with to approximate . Give your answer to six decimal places.

A) and
B) Use Euler's method with to approximate . Give your answer to six decimal places.
B).
C) Solve the initial value problem and estimate the error in parts

Question 40

Match the differential equation with its slope field
(A) (B) (C) i) ii) iii)

Question 41

Twenty-five rabbits were brought to a zoo 15 years ago.
At present there are 35 rabbits in the zoo. The zoo can support a maximum of 180 rabbits.
Assuming a logistic growth model, when will the rabbit population reach 50, 100, and 180 rabbits?

Question 42

A rat population in a certain field is initially 500. After 3 years, the population increases to 700. Assuming logistic growth with a carrying capacity of 1000, how long after reaching 700 will it take for the population to reach 800?

Question 43

A deer population for a certain area is initially . After 6 years, the population increases to 250. After another 4 years, the population increases to 300. Assuming logistic growth with a carrying capacity of 400, what is ?

Question 44

Solve the initial value problem .

Question 45

The differential equation is:

A) linear.
B) nonlinear, but it becomes linear if is the independent variable.
C) separable.
D) nonseparable, but it becomes separable if is the independent variable.
E) none of the above.

Question 46

A deer population for a certain area was 400 at the beginning of the year 1995. After 2 years, the population increased to 500. Assuming logistic growth with a carrying capacity of 800, express the population as a function of t, the number of years since the beginning of 1995.

Question 47

A deer population for a certain area is initially 350. After 2 years, the population increases to 650. Assuming logistic growth with a carrying capacity of 1200, how long after reaching 650 will it take the population to reach 1000?

Question 48

A deer population for a certain area is initially 500. After 3 years, the population increases to 600. Assuming logistic growth with a carrying capacity of 850, what is the deer population 2 years after the population reached 600?

Question 49

Thirty birds were brought to a zoo 10 years ago.
At present, there are 60 birds in the zoo. The zoo can support a maximum of 120 birds.
Assuming a logistic growth model, when will the bird population reach 80, 100, and 120 birds?

Question 50

Solve the linear equations.

A)
B) (Hint: rewrite for .)

Question 51

Twenty panda bears were brought to a national park 20 years ago. At present, there are 42 bears in the park. The park can support a maximum of 200 bears. Assuming a logistic growth model, when will the bear population reach 80, 150, and 200 bears?

Question 52

Let be a population of insects that is modeled by the logistic equation:

A) Determine the equilibrium solutions and their stabilities.
B) What is the long term behavior of the population?
C) Find the value of at the inflection point and explain its meaning referring to the population growth.
D) Solve the logistic equation and verify your answer to B.

Question 53

Let be the population of a certain animal species, satisfying the logistic equation

A) Determine the equilibrium solutions and their stabilities.
B) What is the long term behavior of the population?
C) Find the value of at the inflection point and explain its meaning referring to the population growth.
D) Solve the logistic equation and verify your answer to B.

Question 54

Solve the following differential equations.

A)
B) (Hint: rewrite for .)

Question 55

A rat population for a certain field is 300 at the beginning of the year 2002.. After 4 years, the population increased to 700. Assuming logistic growth with a carrying capacity of 900, express the population as a function of t, the number of years since the beginning of 2002.

Question 56

A rat population for a certain field is initially . After 3 years, the population increases to 450. After another 2 years, the population increases to 650. Assuming logistic growth with a carrying capacity of 900, what is ?

Question 57

A rat population in a certain field is initially 200. After 4 years, the population increases to 350. Assuming logistic growth with a carrying capacity of 550, what is the rat population 3 years after the population reached 350?

Question 58

Bees in a certain region are born at a rate that is proportional to their current population. Without any outside factors the population doubles in 3 weeks' time. It was observed that each day 12 bees joined the population, 10 were caught by men, and 5 died of natural causes.
Determine whether the population will survive if initially it counted 100 bees. If not, when will it die out?

Question 59

A 1200-gallon tank initially contains 500 gallons of water with 4 lbs of salt dissolved in it. Water with a salt concentration of 2 lbs/gal enters the tank at a rate of 10 gal/h, while the solution leaves the tank at a rate of 6 gal/h. Find the amount of salt in the tank when it overflows.

Question 60

Solving the differential equation , we obtain the solution:

A) .
B) .
C) .
D) .
E) .

Question 61

Deer in a certain region, are born at a rate that is proportional to their current population. With the absence of outside factors, the population will double in 3 years' time.
Each year 5 deer join the population, 10 are caught by hunters, and 4 die of natural causes.
If initially there are 50 deer, will the population survive? If not, when will it die out?

Question 62

The curves orthogonal to the solutions of are:

A) .
B) .
C) .
D) .
E) none of the above.

Question 63

Solve the differential equation .

Question 64

Solve the following equations.

A)
B)

Question 65

The differential equation can be solved by which of the following?

A) separating the variables
B) rewriting the equation with as the dependent variable and solving the linear equation
C) rewriting the equation with as the dependent variable and separating the variables
D) Euler's Method
E) none of the above

Question 66

The differential equation can be solved by which of the following?

A) separating the variables
B) rewriting the equation with as the dependent variable and solving the linear equation
C) rewriting the equation with as the dependent variable and separating the variables
D) answers B and C are correct.
E) Euler's method.

Question 67

The differential equation can be solved by which of the following?

A) solving the linear equation - the equation is in the form of a linear equation
B) separating the variables
C) rewriting it with as the dependent variable and solving the resulting linear equation
D) rewriting it with as the dependent variable and solving the resulting separable equation
E) none of the above

Question 68

Solve the initial value problem .

Question 69

Solve the initial value problem .

Question 70

Use the following steps.
A) Write an equation for the areas using integrals.
B) Differentiate the equation in A and solve the resulting linear equation.
Ans:

A) B)
A pool contains 12 million gal of fresh water. Contaminated water flows into the pool at a rate of 4 million gal/year as the mixture in the pool flows out at the same rate.
The concentration of the chemical in the incoming contaminated water is g/gal, where t is in years.
Find the amount of the chemical in the pool at time .

Question 71

The curves orthogonal to the solutions of are:

A) .
B) .
C) .
D) .
E) none of the above.

Question 72

Solve the differential equation .

Question 73

Solve the initial value problem .