Question 1

Solve the equation using an integrating factor. 
-$y ^ { \prime }$ + $t ^ { 2 }$ y = 4 $t ^ { 2 }$ , t > 0

Accepted Answer

A)  y = 4x + C $\mathrm { e } ^ { - \mathrm { t } ^ { 2 } / 2 }$ 
B)  y = 4x + C $e ^ { - t ^ { 3 } / 3 }$ 
C)  y = 4 + C $\mathrm { e } ^ { - \mathrm { t } ^ { 2 } / 2 }$ 
D)  y = 4 + C $e ^ { - t ^ { 3 } / 3 }$ 
A)  y = 4x + C $\mathrm { e } ^ { - \mathrm { t } ^ { 2 } / 2 }$ 
B)  y = 4x + C $e ^ { - t ^ { 3 } / 3 }$ 
C)  y = 4 + C $\mathrm { e } ^ { - \mathrm { t } ^ { 2 } / 2 }$ 
D)  y = 4 + C $e ^ { - t ^ { 3 } / 3 }$

Question 2

Consider the differential equation $y ^ { \prime }$ = g(y) where g(y) is the function whose graph is shown below:  
Indicate whether the following statements are true or false.
-y = 2 is the only constant solution of $y ^ { \prime }$ = g(y) .

Accepted Answer

A) True 
 B)False

Question 3

Given the differential equation: $( t + 1 ) y ^ { \prime } = y t ^ { 2 } - y$ is this the solution $y = c \mathrm { et } ^ { 2 } / 2 ?$

Accepted Answer

A) True 
 B)False

Question 4

One or more initial conditions are given for the differential equation. Use the qualitative theory of autonomous differential equations to sketch the graphs of the corresponding solution. Include a yz-graph as well as a ty-graph. Do these graphs represent: $y ^ { \prime } = 6 - 3 y ; y ( 0 ) = - 1 ; y ( 0 ) = 3 \text { ? }$

Accepted Answer

A) True 
 B)False

Question 5

Solve the equation using an integrating factor. 
-$y ^ { \prime } - 2 y = - 2 t$ , t > 0

Accepted Answer

A)  y = t + 1/2 + C $e ^ { 2 t }$ 
B)  y = t - C $e ^ { 2 t }$ 
C)  y = Ct + $e ^ { 2 t }$ 
D)  y = 1/2 + C $e ^ { - 2 t }$ 
A)  y = t + 1/2 + C $e ^ { 2 t }$ 
B)  y = t - C $e ^ { 2 t }$ 
C)  y = Ct + $e ^ { 2 t }$ 
D)  y = 1/2 + C $e ^ { - 2 t }$

Question 6

Suppose $y ^ { \prime }$ = ky + b and the graphs of several solutions of the differential equation are as below:   Then (I) k is negative.
(II) k is positive.
(III) b is positive.
(IV) b is negative.

Accepted Answer

A)  II and IV 
B)  I and IV 
C)  II and III 
D)  I and III 
E)  not enough information given 
A)  II and IV 
B)  I and IV 
C)  II and III 
D)  I and III 
E)  not enough information given

Question 7

Given the differential equation: $y ^ { \prime } = y ^ { 2 } - t ^ { 2 } y ^ { 2 }$ , is this the solution: $y = \frac { 1 } { \frac { 1 } { 3 } t ^ { 3 } - t + C } ?$

Accepted Answer

A) True 
 B)False

Question 8

Find f'(1) if f(t) is a solution to the initial value problem: $y ^ { \prime } = e ^ { 2 t } - y , y ( 1 ) = 0$ Enter just a real number (no approximations).

Accepted Answer

The answer of Find f'(1) if f(t) is a solution...

Question 9

Given the differential equation with the given initial condition: $y ^ { \prime } = \frac { 3 t ^ { 2 } + 1 } { 2 y } ; \quad y ( 1 ) = - 5$ is this the solution $y = - \sqrt { t ^ { 3 } + t + 2 } ?$

Accepted Answer

A) True 
 B)False

Question 10

Suppose the following is a graph of z = g(y).   Which of the following can then be said about the solution y = f(t) to the initial value problem $y ^ { \prime } = g ( y ) ; \quad y ( 0 ) = - 1$ ? 
(I) f(t) is an increasing function
(II) f(t) is always positive
(III) f(t) has an inflection point when y = 2.

Accepted Answer

A)  III only 
B)  I, II, and III 
C)  I only 
D)  II only 
E)  I and III 
A)  III only 
B)  I, II, and III 
C)  I only 
D)  II only 
E)  I and III

Question 11

Use Euler's method with n = 4 to approximate the solution f(t) to $y ^ { \prime }$ = $y ^ { 2 }$ + ty - 3, $y ( 1 ) = 2$ for $1 \leq t \leq 3$ . Estimate f(3).
Enter just a real number rounded off to two decimal places.

Accepted Answer

The answer of Use Euler's method with n = 4...

Question 12

A patient is receiving a steady infusion of glucose. Let y denote the concentration of glucose in the blood at time t , measured in milligrams of glucose per 100 cubic centimeters of blood, and suppose that y satisfies the differential equation $y ^ { \prime } = 48 - 0.4 y$ . What will be the approximate concentration of glucose in the blood after a long period of time, provided the glucose infusion is continued at the same rate?
Enter just an integer representing the number of mg glucose per 100cc blood (no units)

Accepted Answer

The answer of A patient is receiving a steady infusion...

Question 13

Solve the equation using an integrating factor. 
-$y ^ { \prime } + \frac { 3 } { x } y = 6 x ^ { 2 }$ , t > 0

Accepted Answer

A)  y = $x ^ { 3 }$ + C $x ^ { - 3 }$ 
B)  y = C $x ^ { - 3 }$ 
C)  y = $\mathrm { Cx } ^ { 3 }$ + $x ^ { - 3 }$ 
D)  y = $x ^ { 3 }$ + C 
A)  y = $x ^ { 3 }$ + C $x ^ { - 3 }$ 
B)  y = C $x ^ { - 3 }$ 
C)  y = $\mathrm { Cx } ^ { 3 }$ + $x ^ { - 3 }$ 
D)  y = $x ^ { 3 }$ + C

Question 14

Solve the differential equation with the given initial condition. 
-$y ^ { \prime } = - e ^ { - y }$ , y(0) = 0

Accepted Answer

A)  y = -ln $| t + 1 |$ 
B)  y = 0 
C)  y = $\mathrm { e } ^ { - 1 }$ 
D)  y = $\mathrm { e } ^ { \mathrm { t } }$ 
E)  none of these 
A)  y = -ln $| t + 1 |$ 
B)  y = 0 
C)  y = $\mathrm { e } ^ { - 1 }$ 
D)  y = $\mathrm { e } ^ { \mathrm { t } }$ 
E)  none of these

Question 15

Suppose that $1000 is deposited in a savings account that pays 6% annual interest compounded continuously. At what rate (in dollars per year) is it earning interest after 5 years? Enter just an integer representing the amount to the nearest dollar (no units).

Accepted Answer

The answer of Suppose that $1000 is deposited in a...

Question 16

Given $y ^ { \prime }$ = y(y + 3). On a ty-coordinate system sketch the solutions corresponding to the initial conditions $y ( 0 ) = - 4$ , $y ( 0 ) = - 1$ , and $y ( 0 ) = 1$ . Does this graph represent the situation?

Accepted Answer

A) True 
 B)False

Question 17

Consider the differential equation $y ^ { \prime }$ = g(y) where g(y) is the function whose graph is shown below:  
Indicate whether the following statements are true or false.
-If the initial value of y(0) is 3, then the corresponding solution has an inflection point.

Accepted Answer

A) True 
 B)False

Question 18

Suppose an infectious disease spreads through an elementary school at a rate proportional to the product of the percentage of pupils who have the disease and the percentage of pupils who have not yet contracted the disease. Suppose that at the beginning of the epidemic 5% of the pupils have the disease. Let f(t) be the percentage of pupils who have the disease at time t; give the differential equation satisfied by f(t). Does the following accurately describe this situation: $y ^ { \prime }$ = ky(100 - y); y(0) = 0.05, where k is a positive constant?

Accepted Answer

A) True 
 B)False

Question 19

Find f'(0) if f(t) is a solution to the initial value problem: $y ^ { \prime } = e ^ { 2 t } + y , y ( 0 ) = - 1$ Enter just an integer.

Accepted Answer

The answer of Find f'(0) if f(t) is a solution...

Question 20

After a baby whale is born, its weight gain at any time is proportional to the product of its weight and the difference between its weight and its weight at maturity. Give a differential equation satisfied by f(t), its weight at time t. Does the following accurately describe this situation? 
$\begin{array} { l } 
y ^ { \prime } = k y ( M - y ) \
M = \text { weight at maturit } \
k > 0
\end{array}$

Accepted Answer

A) True 
 B)False

Solve the equation using an integrating factor. - $y ^ { \prime }$ + $t ^ { 2 }$ y = 4 $t ^ { 2 }$ , t > 0

Given the differential equation: $( t + 1 ) y ^ { \prime } = y t ^ { 2 } - y$ is this the solution $y = c \mathrm { et } ^ { 2 } / 2 ?$

Solve the equation using an integrating factor. - $y ^ { \prime } - 2 y = - 2 t$ , t > 0

Given the differential equation: $y ^ { \prime } = y ^ { 2 } - t ^ { 2 } y ^ { 2 }$ , is this the solution: $y = \frac { 1 } { \frac { 1 } { 3 } t ^ { 3 } - t + C } ?$

Find f'(1) if f(t) is a solution to the initial value problem: $y ^ { \prime } = e ^ { 2 t } - y , y ( 1 ) = 0$ Enter just a real number (no approximations).

Given the differential equation with the given initial condition: $y ^ { \prime } = \frac { 3 t ^ { 2 } + 1 } { 2 y } ; \quad y ( 1 ) = - 5$ is this the solution $y = - \sqrt { t ^ { 3 } + t + 2 } ?$

Use Euler's method with n = 4 to approximate the solution f(t) to $y ^ { \prime }$ = $y ^ { 2 }$ + ty - 3, $y ( 1 ) = 2$ for $1 \leq t \leq 3$ . Estimate f(3). Enter just a real number rounded off to two decimal places.

Solve the equation using an integrating factor. - $y ^ { \prime } + \frac { 3 } { x } y = 6 x ^ { 2 }$ , t > 0

Solve the differential equation with the given initial condition. - $y ^ { \prime } = - e ^ { - y }$ , y(0) = 0

Suppose that $1000 is deposited in a savings account that pays 6% annual interest compounded continuously. At what rate (in dollars per year) is it earning interest after 5 years? Enter just an integer representing the amount to the nearest dollar (no units).

Find f'(0) if f(t) is a solution to the initial value problem: $y ^ { \prime } = e ^ { 2 t } + y , y ( 0 ) = - 1$ Enter just an integer.

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Exam 10: Differential Equations

Solve the equation using an integrating factor. - y′y ^ { \prime }y′ + t2t ^ { 2 }t2 y = 4 t2t ^ { 2 }t2 , t > 0

Consider the differential equation y′y ^ { \prime }y′ = g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false. -y = 2 is the only constant solution of y′y ^ { \prime }y′ = g(y) .

Given the differential equation: (t+1)y′=yt2−y( t + 1 ) y ^ { \prime } = y t ^ { 2 } - y(t+1)y′=yt2−y is this the solution y=cet2/2?y = c \mathrm { et } ^ { 2 } / 2 ?y=cet2/2?

Solve the equation using an integrating factor. - y′−2y=−2ty ^ { \prime } - 2 y = - 2 ty′−2y=−2t , t > 0

Suppose y′y ^ { \prime }y′ = ky + b and the graphs of several solutions of the differential equation are as below: Then (I) k is negative. (II) k is positive. (III) b is positive. (IV) b is negative.

Given the differential equation: y′=y2−t2y2y ^ { \prime } = y ^ { 2 } - t ^ { 2 } y ^ { 2 }y′=y2−t2y2 , is this the solution: y=113t3−t+C?y = \frac { 1 } { \frac { 1 } { 3 } t ^ { 3 } - t + C } ?y=31​t3−t+C1​?

Find f'(1) if f(t) is a solution to the initial value problem: y′=e2t−y,y(1)=0y ^ { \prime } = e ^ { 2 t } - y , y ( 1 ) = 0y′=e2t−y,y(1)=0 Enter just a real number (no approximations).

Given the differential equation with the given initial condition: y′=3t2+12y;y(1)=−5y ^ { \prime } = \frac { 3 t ^ { 2 } + 1 } { 2 y } ; \quad y ( 1 ) = - 5y′=2y3t2+1​;y(1)=−5 is this the solution y=−t3+t+2?y = - \sqrt { t ^ { 3 } + t + 2 } ?y=−t3+t+2​?

Use Euler's method with n = 4 to approximate the solution f(t) to y′y ^ { \prime }y′ = y2y ^ { 2 }y2 + ty - 3, y(1)=2y ( 1 ) = 2y(1)=2 for 1≤t≤31 \leq t \leq 31≤t≤3 . Estimate f(3). Enter just a real number rounded off to two decimal places.

Solve the equation using an integrating factor. - y′+3xy=6x2y ^ { \prime } + \frac { 3 } { x } y = 6 x ^ { 2 }y′+x3​y=6x2 , t > 0

Solve the differential equation with the given initial condition. - y′=−e−yy ^ { \prime } = - e ^ { - y }y′=−e−y , y(0) = 0

Suppose that $1000 is deposited in a savings account that pays 6% annual interest compounded continuously. At what rate (in dollars per year) is it earning interest after 5 years? Enter just an integer representing the amount to the nearest dollar (no units).

Given y′y ^ { \prime }y′ = y(y + 3). On a ty-coordinate system sketch the solutions corresponding to the initial conditions y(0)=−4y ( 0 ) = - 4y(0)=−4 , y(0)=−1y ( 0 ) = - 1y(0)=−1 , and y(0)=1y ( 0 ) = 1y(0)=1 . Does this graph represent the situation?

Consider the differential equation y′y ^ { \prime }y′ = g(y) where g(y) is the function whose graph is shown below: Indicate whether the following statements are true or false. -If the initial value of y(0) is 3, then the corresponding solution has an inflection point.

Find f'(0) if f(t) is a solution to the initial value problem: y′=e2t+y,y(0)=−1y ^ { \prime } = e ^ { 2 t } + y , y ( 0 ) = - 1y′=e2t+y,y(0)=−1 Enter just an integer.

The Derivative

Applications of the Derivative

Techniques of Differentiation

Logarithm Functions

Applications of the Exponential and Natural Logarithm Functions

The Definite Integral

Functions of Several Variables

The Trigonometric Functions

Techniques of Integration

Taylor Polynomials and Infinite Series

Probability and Calculus

Filters

Solve the equation using an integrating factor. - $y ^ { \prime }$ + $t ^ { 2 }$ y = 4 $t ^ { 2 }$ , t > 0

Given the differential equation: $( t + 1 ) y ^ { \prime } = y t ^ { 2 } - y$ is this the solution $y = c \mathrm { et } ^ { 2 } / 2 ?$

Solve the equation using an integrating factor. - $y ^ { \prime } - 2 y = - 2 t$ , t > 0

Given the differential equation: $y ^ { \prime } = y ^ { 2 } - t ^ { 2 } y ^ { 2 }$ , is this the solution: $y = \frac { 1 } { \frac { 1 } { 3 } t ^ { 3 } - t + C } ?$

Find f'(1) if f(t) is a solution to the initial value problem: $y ^ { \prime } = e ^ { 2 t } - y , y ( 1 ) = 0$ Enter just a real number (no approximations).

Given the differential equation with the given initial condition: $y ^ { \prime } = \frac { 3 t ^ { 2 } + 1 } { 2 y } ; \quad y ( 1 ) = - 5$ is this the solution $y = - \sqrt { t ^ { 3 } + t + 2 } ?$

Use Euler's method with n = 4 to approximate the solution f(t) to $y ^ { \prime }$ = $y ^ { 2 }$ + ty - 3, $y ( 1 ) = 2$ for $1 \leq t \leq 3$ . Estimate f(3). Enter just a real number rounded off to two decimal places.

Solve the equation using an integrating factor. - $y ^ { \prime } + \frac { 3 } { x } y = 6 x ^ { 2 }$ , t > 0

Solve the differential equation with the given initial condition. - $y ^ { \prime } = - e ^ { - y }$ , y(0) = 0

Find f'(0) if f(t) is a solution to the initial value problem: $y ^ { \prime } = e ^ { 2 t } + y , y ( 0 ) = - 1$ Enter just an integer.