Question 1

Consider the initial value problem
   
Find all solutions of this initial value problem.

Accepted Answer

The answer of Consider the initial value problem
  ...

Question 2

What is the general solution of the differential equation

Accepted Answer

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

Question 3

What is the solution of this initial value problem?

Accepted Answer

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

Question 4

Consider the difference equation   = 0.50   + 6, n = 0, 1, 2, 3...Find the explicit solution of this difference equation in terms of   .

Accepted Answer

A)  $ y_{n}=0.50^{n} y_{0}+12, n=0,1,2,3 \ldots $ 
B)  $ y_{n}=0.50^{n} y_{0}+4, n=0,1,2,3 \ldots $ 
C)  $ y_{n}=0.50^{n-1} y_{0}+6, n=0,1,2,3 \ldots $ 
D)  $ y_{n}=0.50^{n-1} y_{0}+12, n=0,1,2,3 \ldots $ 
A)  $ y_{n}=0.50^{n} y_{0}+12, n=0,1,2,3 \ldots $ 
B)  $ y_{n}=0.50^{n} y_{0}+4, n=0,1,2,3 \ldots $ 
C)  $ y_{n}=0.50^{n-1} y_{0}+6, n=0,1,2,3 \ldots $ 
D)  $ y_{n}=0.50^{n-1} y_{0}+12, n=0,1,2,3 \ldots $

Question 5

What is the general solution of the differential equation

Accepted Answer

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

Question 6

Consider the differential equation  
Which of these is the general solution to the equation?

Accepted Answer

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

Question 7

Identify the integrating factor for this linear differential equation: 
  - 2y =

Accepted Answer

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

Question 8

Consider the difference equation   
Find the explicit solution of this difference equation in terms of y0 .

Accepted Answer

A) $ y_{n}=-0.55 n y_{0}, n=0,1,2,3 \ldots $ 
B)  $ y_{n}=(-0.55)^{n} y_{0}, n=0,1,2,3 \ldots $ 
C) $ y_{n}=-0.55^{n} y_{0}, n=0,1,2,3 \ldots $ 
D)  $ y_{n}=\left(-0.55 y_{0}\right)^{n}, n=0,1,2,3 $ 
A) $ y_{n}=-0.55 n y_{0}, n=0,1,2,3 \ldots $ 
B)  $ y_{n}=(-0.55)^{n} y_{0}, n=0,1,2,3 \ldots $ 
C) $ y_{n}=-0.55^{n} y_{0}, n=0,1,2,3 \ldots $ 
D)  $ y_{n}=\left(-0.55 y_{0}\right)^{n}, n=0,1,2,3 $

Question 9

Consider the differential equation 
   
(i) Which of these is the general solution of this equation?
   
(ii) What choice of the arbitrary constant in the general solution ensures that the solution curve passes through the point

Accepted Answer

The answer of Consider the differential equation 
  ...

Question 10

Solve this initial value problem:

Accepted Answer

A)  $ 6 x^{2}-x y+8 y^{2}=0 $ 
B)  $ 6 x^{2}+x y+8 y^{2}=0 $ 
C)  $ 3 x^{2}-x y+4 y^{2}=0 $ 
D)  $ 3 x^{2}+x y+4 y^{2}=0 $ 
A)  $ 6 x^{2}-x y+8 y^{2}=0 $ 
B)  $ 6 x^{2}+x y+8 y^{2}=0 $ 
C)  $ 3 x^{2}-x y+4 y^{2}=0 $ 
D)  $ 3 x^{2}+x y+4 y^{2}=0 $

Question 11

Consider the following initial value problem:
   
(i) Use Euler's method with two equal steps to find an approximation of y(1). Enter the exact answer, not an approximation.
(ii) Solve the initial value problem and compute the solution at x = 1. Enter the exact answer, not an approximation.
(iii) What is the error in Euler's method in making this approximation?

Accepted Answer

The answer of Consider the following initial value problem:
 ...

Question 12

What is the general solution of the differential equation (1 +   )   = 1 +   ?

Accepted Answer

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

Question 13

Which of these is the general solution of this homogeneous differential equation?

Accepted Answer

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

Question 14

A city's water reservoir contains 6 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 0.25 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.If the purification system fails, how much time (in days) elapses before the alarm is triggered? Round your answer to the nearest hundredth of a day.

Accepted Answer

The answer of A city's water reservoir contains 6 billion...

Question 15

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 0.6 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 0.25 bcm per day, and the concentration of pollutants in the inflow is 2 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. Let t = 0 be the time when the purification system fails. What is x(0)?

Accepted Answer

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

Question 16

A model of a fishery which grows logistically and is harvested at a constant rate is given by
 
Which of these is an equilibrium solution of this differential equation? Select all that apply.

Accepted Answer

A)  y = 0 
B)  y = -1 
C)  y = 6 
D)  y = 1 
E)  y = -6 
A)  y = 0 
B)  y = -1 
C)  y = 6 
D)  y = 1 
E)  y = -6

Question 17

Consider the differential equation   
Which of the following is the general solution of this equation?

Accepted Answer

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

Question 18

Consider the difference equation  
Which of the following is an accurate description of the behavior of the solutions to this difference equation in terms of y0 . ?

Accepted Answer

A)  If y0 (\neq\) 0, then the sequence {   } diverges to $\infty$ as n $\rightarrow$ $\infty$. 
B)  The sequence {   } converges to 6.25 as n $\rightarrow$ $\infty$ for all values of . 
C)  If y0 > 0, then the sequence {   } converges to 6.25 as n$\rightarrow$ $\infty$; for all other choices of y0 , the sequence {   } converges to 0 as n $\rightarrow$ $\infty$. 
D)  
D)  If y0 $\neq$0, then the sequence {   } converges to 6.25 as n $\rightarrow$ $\infty$; for all other choices of y0 , the sequence {   } converges to 0 as n $\rightarrow$ $\infty$. 
A)  If y0 (\neq\) 0, then the sequence {   } diverges to $\infty$ as n $\rightarrow$ $\infty$. 
B)  The sequence {   } converges to 6.25 as n $\rightarrow$ $\infty$ for all values of . 
C)  If y0 > 0, then the sequence {   } converges to 6.25 as n$\rightarrow$ $\infty$; for all other choices of y0 , the sequence {   } converges to 0 as n $\rightarrow$ $\infty$. 
D)  
D)  If y0 $\neq$0, then the sequence {   } converges to 6.25 as n $\rightarrow$ $\infty$; for all other choices of y0 , the sequence {   } converges to 0 as n $\rightarrow$ $\infty$.

Question 19

Consider the autonomous differential equation
 
Which of these is an equilibrium solution of this differential equation? Select all that apply.

Accepted Answer

A)  y = -5 
B)  y = 5 
C)  y = 0 
D)  y = -10 
E)  y = 10 
A)  y = -5 
B)  y = 5 
C)  y = 0 
D)  y = -10 
E)  y = 10

Question 20

Consider the autonomous differential equation 
   
Determine   for the initial condition y(t0 ) = (8, -4).

Accepted Answer

Consider the initial value problem Find all solutions of this initial value problem.

What is the general solution of the differential equation

What is the solution of this initial value problem?

Consider the difference equation = 0.50 + 6, n = 0, 1, 2, 3...Find the explicit solution of this difference equation in terms of .

What is the general solution of the differential equation

Consider the differential equation Which of these is the general solution to the equation?

Identify the integrating factor for this linear differential equation: - 2y =

Consider the difference equation Find the explicit solution of this difference equation in terms of y₀ .

Consider the differential equation (i) Which of these is the general solution of this equation? (ii) What choice of the arbitrary constant in the general solution ensures that the solution curve passes through the point

Solve this initial value problem:

What is the general solution of the differential equation (1 + ) = 1 + ?

Which of these is the general solution of this homogeneous differential equation?

A model of a fishery which grows logistically and is harvested at a constant rate is given by Which of these is an equilibrium solution of this differential equation? Select all that apply.

Consider the differential equation Which of the following is the general solution of this equation?

Consider the difference equation Which of the following is an accurate description of the behavior of the solutions to this difference equation in terms of y₀ . ?

Consider the autonomous differential equation Which of these is an equilibrium solution of this differential equation? Select all that apply.

Consider the autonomous differential equation Determine for the initial condition y(t₀ ) = (8, -4).

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

Consider the initial value problem Find all solutions of this initial value problem.

What is the general solution of the differential equation

What is the solution of this initial value problem?

Consider the difference equation = 0.50 + 6, n = 0, 1, 2, 3...Find the explicit solution of this difference equation in terms of .

What is the general solution of the differential equation

Consider the differential equation Which of these is the general solution to the equation?

Identify the integrating factor for this linear differential equation: - 2y =

Consider the difference equation Find the explicit solution of this difference equation in terms of y0 .

Consider the differential equation (i) Which of these is the general solution of this equation? (ii) What choice of the arbitrary constant in the general solution ensures that the solution curve passes through the point

Solve this initial value problem:

What is the general solution of the differential equation (1 + ) = 1 + ?

Which of these is the general solution of this homogeneous differential equation?

A model of a fishery which grows logistically and is harvested at a constant rate is given by Which of these is an equilibrium solution of this differential equation? Select all that apply.

Consider the differential equation Which of the following is the general solution of this equation?

Consider the difference equation Which of the following is an accurate description of the behavior of the solutions to this difference equation in terms of y0 . ?

Consider the autonomous differential equation Which of these is an equilibrium solution of this differential equation? Select all that apply.

Consider the autonomous differential equation Determine for the initial condition y(t0 ) = (8, -4).

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

Consider the difference equation Find the explicit solution of this difference equation in terms of y₀ .

Consider the difference equation Which of the following is an accurate description of the behavior of the solutions to this difference equation in terms of y₀ . ?

Consider the autonomous differential equation Determine for the initial condition y(t₀ ) = (8, -4).