Question 1

What is the general solution of the differential equation

Accepted Answer

A)   
B)   
C)   
D)   
A)   
B)   
C)   
D)

Question 2

A pie is moved from the oven at 475 degrees Fahrenheit to a freezer at 25 degrees Fahrenheit. After 10 minutes, the pie has cooled to 425 degrees Fahrenheit.
Let T(t) be the temperature of the pie t minutes after it has been moved to the freezer. Formulate an initial-value problem whose solution is T(t).

Accepted Answer

@#IMG-DLM& where k is a negati

Question 3

Consider the differential equation   
.What choice of the arbitrary constant in the general solution ensures that the solution curve passes through the point (1, 4)?

Accepted Answer

The answer of Consider the differential equation   
.What...

Question 4

Which of the following is an accurate conclusion that can be made using the existence and uniqueness theorem for first-order nonlinear equations for this initial value problem?

Accepted Answer

A)  The initial value problem has a unique solution because f (x, y) is continuous on a rectangle containing the point (2, 8) on its boundary. 
B)  The initial value problem is not guaranteed to have a unique solution because fx (x, y) is not continuous when x = -1. 
C)  The initial value problem has a unique solution because both f (x, y) and fy (x, y) are continuous on a rectangle containing the point (2, 8). 
D)  The initial value problem is not guaranteed to have a unique local solution because there is no rectangle surrounding the point (2, 8) on which both f (x, y) and fy(x, y) are continuous. 
A)  The initial value problem has a unique solution because f (x, y) is continuous on a rectangle containing the point (2, 8) on its boundary. 
B)  The initial value problem is not guaranteed to have a unique solution because fx (x, y) is not continuous when x = -1. 
C)  The initial value problem has a unique solution because both f (x, y) and fy (x, y) are continuous on a rectangle containing the point (2, 8). 
D)  The initial value problem is not guaranteed to have a unique local solution because there is no rectangle surrounding the point (2, 8) on which both f (x, y) and fy(x, y) are continuous.

Question 5

A city's water reservoir contains 7 billion cubic meters (bcm) of water. The purification system ensures that the concentration of pollutants remains constant at 0.5 kilograms per bcm, and sensors will trigger an alarm if the concentration of pollutants rises above 1 kilogram per bcm. Water flows in and out of the reservoir at the same rate of 0.20 bcm per day, and the concentration of pollutants in the inflow is 1.9 kilograms per bcm. At all times, the reservoir is well mixed.Set up a differential equation whose solution x(t) is the amount of pollutant in the reservoir at time t.
   ________________

Accepted Answer

The answer of A city's water reservoir contains 7 billion...

Question 6

Consider the differential equation  
 What is the general solution of this differential equation?

Accepted Answer

A)  $ x^{4} y^{8}+\frac{e^{2 x}}{2}+\frac{\cos \left(\frac{1}{3} y\right)}{\frac{1}{3}}=C $ 
B)  $ x^{4} y^{8}+\frac{e^{2 x}}{2}-\frac{\cos \left(\frac{1}{3} y\right)}{\frac{1}{3}}=C $ 
C)  $ x^{4} y^{8}+2 e^{2 x}-\frac{1}{3} \cos \left(\frac{1}{3} v\right)=C $ 
D)  $ x^{4} y^{8}+2 e^{2 x}+\frac{1}{3} \cos \left(\frac{1}{3} v\right)=C $ 
A)  $ x^{4} y^{8}+\frac{e^{2 x}}{2}+\frac{\cos \left(\frac{1}{3} y\right)}{\frac{1}{3}}=C $ 
B)  $ x^{4} y^{8}+\frac{e^{2 x}}{2}-\frac{\cos \left(\frac{1}{3} y\right)}{\frac{1}{3}}=C $ 
C)  $ x^{4} y^{8}+2 e^{2 x}-\frac{1}{3} \cos \left(\frac{1}{3} v\right)=C $ 
D)  $ x^{4} y^{8}+2 e^{2 x}+\frac{1}{3} \cos \left(\frac{1}{3} v\right)=C $

Question 7

What is the two-parameter family of solutions of the second-order differential equation

Accepted Answer

A)  $ y=\frac{21}{16} x^{2}+C_{1} x^{\frac{6}{7}}+C_{2} $ 
B)  $ y=C_{1} x^{2}+C_{1} x^{-6} $ 
C)  $ y=C_{1} x^{2}+C_{1} x^{\frac{6}{7}} $ 
D)  $ y=\frac{21}{16} x^{2}+C_{1}+C_{2} x^{-6} $ 
A)  $ y=\frac{21}{16} x^{2}+C_{1} x^{\frac{6}{7}}+C_{2} $ 
B)  $ y=C_{1} x^{2}+C_{1} x^{-6} $ 
C)  $ y=C_{1} x^{2}+C_{1} x^{\frac{6}{7}} $ 
D)  $ y=\frac{21}{16} x^{2}+C_{1}+C_{2} x^{-6} $

Question 8

Consider the autonomous differential equation
   
dentify the following statement as TRUE or FALSE:A solution curve passing through the point (0, -2) tends to 0 as t $\rightarrow$$\infty$.

Accepted Answer

A) True 
 B)False

Question 9

Consider the differential equation   Find an integrating factor μ(x) so that the following differential equation is exact:

Accepted Answer

The answer of Consider the differential equation   Find...

Question 10

Identify the integrating factor for this linear differential equation:

Accepted Answer

The answer of Identify the integrating factor for this linear...

Question 11

Find the general solution to the differential equation tan(   x) 
  + y = -5 sin(   x).

Accepted Answer

A)   
B)   
C)   
D)   
A)   
B)   
C)   
D)

Question 12

Which of the following statements are true for this initial-value problem? Select all that apply

Accepted Answer

A)  A locally unique solution is not guaranteed to exist by the local existence and uniqueness theorem for first-order differential equations because   is not continuous at the point . 
B)  y = x - 19 is the only solution of this initial value problem. 
C)  y = x - 19 and y = 1 - x are both solutions of this initial value problem. 
D)  This initial value problem cannot have a solution because the conditions of the existence and uniqueness theorem for first-order linear equations are not satisfied. 
E)  The existence and uniqueness theorem for first-order linear equations ensures the existence of a unique local solution of this initial value problem because x - 10 is continuous at the point (10, 9). 
A)  A locally unique solution is not guaranteed to exist by the local existence and uniqueness theorem for first-order differential equations because   is not continuous at the point . 
B)  y = x - 19 is the only solution of this initial value problem. 
C)  y = x - 19 and y = 1 - x are both solutions of this initial value problem. 
D)  This initial value problem cannot have a solution because the conditions of the existence and uniqueness theorem for first-order linear equations are not satisfied. 
E)  The existence and uniqueness theorem for first-order linear equations ensures the existence of a unique local solution of this initial value problem because x - 10 is continuous at the point (10, 9).

Question 13

Consider the autonomous differential equation
 
Which of the following statements is true?

Accepted Answer

A)  All nonequilibrium solutions tend toward -$\infty$ as t $\rightarrow$$\infty$. 
B)  y = 2k $\pi$ is stable and y = (2k + 1) $\pi$ is unstable, for any integer k. 
C)  y = 2k $\pi$is unstable and y = (2k + 1) $\pi$ is stable, for any integer k. 
D)  All equilibrium solutions are semi-stable. 
A)  All nonequilibrium solutions tend toward -$\infty$ as t $\rightarrow$$\infty$. 
B)  y = 2k $\pi$ is stable and y = (2k + 1) $\pi$ is unstable, for any integer k. 
C)  y = 2k $\pi$is unstable and y = (2k + 1) $\pi$ is stable, for any integer k. 
D)  All equilibrium solutions are semi-stable.

Question 14

Which of the following is an accurate conclusion that can be made using the existence and uniqueness theorem for first-order differential equations for this initial value problem?

Accepted Answer

A)  The initial value problem has a unique solution because f (x, y) is continuous on a rectangle containing the point (10, 6). 
B)  The initial value problem is not guaranteed to have a unique solution because fx (x, y) is not continuous when x = -9. 
C)  The initial value problem has a unique solution because both f (x, y) and fy(x, y) are continuous on a rectangle containing the point (10, 6). 
D)  The initial value problem does not have a solution because fx (x, y) and fy (x, y) are not both continuous on a rectangle containing the point (10, 6). 
A)  The initial value problem has a unique solution because f (x, y) is continuous on a rectangle containing the point (10, 6). 
B)  The initial value problem is not guaranteed to have a unique solution because fx (x, y) is not continuous when x = -9. 
C)  The initial value problem has a unique solution because both f (x, y) and fy(x, y) are continuous on a rectangle containing the point (10, 6). 
D)  The initial value problem does not have a solution because fx (x, y) and fy (x, y) are not both continuous on a rectangle containing the point (10, 6).

Question 15

What is the solution of this initial value problem?

Accepted Answer

A)   
B)   
C)   
D)   
A)   
B)   
C)   
D)

Question 16

A ball is thrown from the top of a tall building with a speed of 12 meters per second. Suppose the ball hits the ground with a speed of 69 meters per second. (Recall that the acceleration due to gravity is 9.8 m/s2.)What is the time   (in seconds) of impact? Round your answer to the nearest hundredth of a second.

Accepted Answer

Question 17

Identify the integrating factor for this linear differential equation:

Accepted Answer

The answer of Identify the integrating factor for this linear...

Question 18

On another planet, a ball dropped from a height of 5 meters takes 3 seconds to hit the ground.
Let g be the acceleration due to gravity on this planet, v0 the initial velocity of the ball, and x0 the initial height of the ball above the ground. Write a formula for the height of the ball, x(t), above the ground at time t.

Accepted Answer

Question 19

Find the general solution of the differential equation x   - 2y =   , x > 0.

Accepted Answer

A)  y = 4   + C   
B)  y = 4   + C   
C)  y =   + C   
D)  y =   + C   
A)  y = 4   + C   
B)  y = 4   + C   
C)  y =   + C   
D)  y =   + C

Question 20

On another planet, a ball dropped from a height of 5 meters takes 4.5 seconds to hit the ground.Find the amount of time it takes the ball to hit the ground if it is dropped from a height of 122 meters. Round your answer to the nearest tenth of a second.

Accepted Answer

The answer of On another planet, a ball dropped from...

What is the general solution of the differential equation

Consider the differential equation .What choice of the arbitrary constant in the general solution ensures that the solution curve passes through the point (1, 4)?

Which of the following is an accurate conclusion that can be made using the existence and uniqueness theorem for first-order nonlinear equations for this initial value problem?

Consider the differential equation What is the general solution of this differential equation?

What is the two-parameter family of solutions of the second-order differential equation

Consider the differential equation Find an integrating factor μ(x) so that the following differential equation is exact:

Identify the integrating factor for this linear differential equation:

Find the general solution to the differential equation tan( x) + y = -5 sin( x).

Which of the following statements are true for this initial-value problem? Select all that apply

Consider the autonomous differential equation Which of the following statements is true?

Which of the following is an accurate conclusion that can be made using the existence and uniqueness theorem for first-order differential equations for this initial value problem?

What is the solution of this initial value problem?

Identify the integrating factor for this linear differential equation:

Find the general solution of the differential equation x - 2y = , x > 0.

On another planet, a ball dropped from a height of 5 meters takes 4.5 seconds to hit the ground.Find the amount of time it takes the ball to hit the ground if it is dropped from a height of 122 meters. Round your answer to the nearest tenth of a second.

Introduction

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters

Exam 2: First-Order Differential Equations

What is the general solution of the differential equation

Consider the differential equation .What choice of the arbitrary constant in the general solution ensures that the solution curve passes through the point (1, 4)?

Which of the following is an accurate conclusion that can be made using the existence and uniqueness theorem for first-order nonlinear equations for this initial value problem?

Consider the differential equation What is the general solution of this differential equation?

What is the two-parameter family of solutions of the second-order differential equation

Consider the autonomous differential equation dentify the following statement as TRUE or FALSE:A solution curve passing through the point (0, -2) tends to 0 as t →\rightarrow→∞\infty∞ .

Consider the differential equation Find an integrating factor μ(x) so that the following differential equation is exact:

Identify the integrating factor for this linear differential equation:

Find the general solution to the differential equation tan( x) + y = -5 sin( x).

Which of the following statements are true for this initial-value problem? Select all that apply

Consider the autonomous differential equation Which of the following statements is true?

Which of the following is an accurate conclusion that can be made using the existence and uniqueness theorem for first-order differential equations for this initial value problem?

What is the solution of this initial value problem?

Identify the integrating factor for this linear differential equation:

Find the general solution of the differential equation x - 2y = , x > 0.

On another planet, a ball dropped from a height of 5 meters takes 4.5 seconds to hit the ground.Find the amount of time it takes the ball to hit the ground if it is dropped from a height of 122 meters. Round your answer to the nearest tenth of a second.

Introduction

Second-Order Linear Differential Equations

Higher-Order Linear Differential Equations

Series Solutions of Second-Order Linear Equations

The Laplace Transform

Systems of First-Order Linear Equations

Numerical Methods

Nonlinear Differential Equations and Stability

Partial Differential Equations and Fourier Series

Boundary Value Problems and Sturm-Liouville Theory

Filters