Question 1

Let f(t) be the solution to of $y ^ { \prime }$ = $y ^ { 2 }$ t - y, y(0) = 1. Which of the following statements is true?

Accepted Answer

A)  f is increasing at the origin. 
B)  f(t) will be a constant solution of the differential equation. 
C)  f is decreasing at the origin. 
D)  $f ^ { \prime }$ (1) = 0 
A)  f is increasing at the origin. 
B)  f(t) will be a constant solution of the differential equation. 
C)  f is decreasing at the origin. 
D)  $f ^ { \prime }$ (1) = 0

Question 2

Given the differential equation with the given initial condition: $y ^ { \prime } = e ^ { - y } ; \quad y ( 0 ) = 0$ is this the solution $y = \ln | t + 1 | ?$

Accepted Answer

A) True 
 B)False

Question 3

Solve the differential equation with the given initial condition. 
-$y ^ { \prime } = \frac { x } { y ^ { \prime } } y ( 3 ) = 5$

Accepted Answer

A)  y = $\sqrt { x ^ { 2 } + 9 }$ 
B)  y = ± $\sqrt { x ^ { 2 } + 16 }$ 
C)  y = ± $\sqrt { x ^ { 2 } + 9 }$ 
D)  y = $\sqrt { x ^ { 2 } + 16 }$ 
A)  y = $\sqrt { x ^ { 2 } + 9 }$ 
B)  y = ± $\sqrt { x ^ { 2 } + 16 }$ 
C)  y = ± $\sqrt { x ^ { 2 } + 9 }$ 
D)  y = $\sqrt { x ^ { 2 } + 16 }$

Question 4

Use Euler's method with n = 4 on the interval $0 \leq t \leq 2$ to approximate the solution f(t) to $y ^ { \prime } = y - 4 t$ , $y ( 0 ) = 2$ Is the following graph accurate?

Accepted Answer

A) True 
 B)False

Question 5

Find a constant solution of $y ^ { \prime } = 10 y - 7$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Find a constant solution of \(y ^...

Question 6

Given the differential equation: $\left( t ^ { 2 } + 1 \right) y ^ { \prime } = y t$ , is this the solution $y = c \sqrt { t ^ { 2 } + 1 } ?$

Accepted Answer

A) True 
 B)False

Question 7

Given the differential equation: y' is this the solution $y = \pm \sqrt { 2 \ln | t | + C } ?$

Accepted Answer

A) True 
 B)False

Question 8

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. $y ^ { \prime } = 6 + 2 y ; y ( 0 ) = - 4 ; y ( 0 ) = - 2$ Do these graphs represent the situation?

Accepted Answer

A) True 
 B)False

Question 9

How much would you need to invest per month - in effect, continuously - in an investment account that pays an annual interest rate of 9%, compounded continuously, in order for the account to be worth $100,000 after 20 years? Enter just an integer representing dollars to the nearest dollar (no units)

Accepted Answer

The answer of How much would you need to invest...

Question 10

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.
-Let $y ^ { \prime }$ = 2 - y. Which of the following properties hold for the solution y = f(t) determined by the initial condition y(0) = 1? (I) It is always concave down.
(II) It is a constant solution.
(III) It is always decreasing.

Accepted Answer

A)  I only 
B)  III only 
C)  I and III 
D)  II only 
E)  none of these 
A)  I only 
B)  III only 
C)  I and III 
D)  II only 
E)  none of these

Question 11

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. 2t $y ^ { \prime }$ - y = $\frac { 6 } { t }$ ; y(1) = -1, t > 0

Accepted Answer

A)  integrating factor:\ $t ^ { - 1 / 2 }$
General solution: y = - $\frac { 2 } { t }$ + C $t ^ {  1 / 2 }$
Particular solution: y = - $\frac { 2 } { t }$ + $t ^ { 1 / 2 }$ 
B)  integrating factor: 2t
General solution: y = - $\frac { 3 } { 4 t ^ { 2 } }$ + $\frac { C } { 2 t }$
Particular solution: y = - $\frac { 3 } { 4 t ^ { 2 } }$ + $\frac { 1 } { 4 t }$ 
C)  integrating factor: $t ^ { - 1 / 2 }$
General solution: y = -4 + C $t ^ {  1 / 2 }$
Particular solution: y = -4 + 3 $t ^ {  1 / 2 }$ 
D)  integrating factor: 2t
General solution: y = - $\frac { 3 } { 2 t ^ { 2 } }$ + $\frac { C } { 2 t }$
Particular solution: y = - $\frac { 3 } { 2 t ^ { 2 } }$ + $\frac { 1 } { 2 t }$ 
A)  integrating factor:\ $t ^ { - 1 / 2 }$
General solution: y = - $\frac { 2 } { t }$ + C $t ^ {  1 / 2 }$
Particular solution: y = - $\frac { 2 } { t }$ + $t ^ { 1 / 2 }$ 
B)  integrating factor: 2t
General solution: y = - $\frac { 3 } { 4 t ^ { 2 } }$ + $\frac { C } { 2 t }$
Particular solution: y = - $\frac { 3 } { 4 t ^ { 2 } }$ + $\frac { 1 } { 4 t }$ 
C)  integrating factor: $t ^ { - 1 / 2 }$
General solution: y = -4 + C $t ^ {  1 / 2 }$
Particular solution: y = -4 + 3 $t ^ {  1 / 2 }$ 
D)  integrating factor: 2t
General solution: y = - $\frac { 3 } { 2 t ^ { 2 } }$ + $\frac { C } { 2 t }$
Particular solution: y = - $\frac { 3 } { 2 t ^ { 2 } }$ + $\frac { 1 } { 2 t }$

Question 12

Solve the differential equation with the given initial condition. 
-$y ^ { \prime } = t y , y ( 0 ) = - 1$

Accepted Answer

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

Question 13

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false:  
-$f ^ { \prime }$ $\left( \frac { 3 } { 2 } \right)$ ≈ $\frac { 3 } { 2 }$

Accepted Answer

A) True 
 B)False

Question 14

Find f'(1) if f(t) is a solution to the initial value problem: $y ^ { \prime } = t y ^ { 2 } + 5 , y ( 1 ) = 1$ Enter just an integer.

Accepted Answer

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

Question 15

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.. $y ^ { \prime } = y ^ { 2 } - 9 ; y ( 0 ) = - 5 ; y ( 0 ) = 2$ Do these graphs represent the situation?

Accepted Answer

A) True 
 B)False

Question 16

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false:  
-f(0) = 1

Accepted Answer

A) True 
 B)False

Question 17

Given the differential equation: $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { \mathrm { t } + \mathrm { e } ^ { \mathrm { t } } } { \mathrm { y } } \text {, }$ is this the solution $y = \pm \sqrt { t ^ { 2 } + 2 e ^ { t } + C } ?$

Accepted Answer

A) True 
 B)False

Question 18

Solve the differential equation with the given initial condition. 
-$y ^ { \prime } = y ( t - 2 ) ; y ( 0 ) = 1$

Accepted Answer

A)  y = $e \left[ \left( t ^ { 2 } / 2 \right) - 2 t \right]$ - 2t + 1 
B)  y = $\frac { t ^ { 2 } } { 2 }$ - 2t 
C)  y = $e \left[ \left( t ^ { 2 } / 2 \right) - 2 t \right]$ 
D)  y = 0 
E)  none of these 
A)  y = $e \left[ \left( t ^ { 2 } / 2 \right) - 2 t \right]$ - 2t + 1 
B)  y = $\frac { t ^ { 2 } } { 2 }$ - 2t 
C)  y = $e \left[ \left( t ^ { 2 } / 2 \right) - 2 t \right]$ 
D)  y = 0 
E)  none of these

Question 19

A millionaire wants to set up a trust for her grandchild. She wants to put a lump sum of money into an account earning 10% interest. She'd like her grandchild to be able to withdraw $100 every month for the rest of the child's life. Write a differential equation satisfied by f(t), the amount of money in the account at time $\text { t. }$ Does the equation, $y ^ { \prime } = 0.1 \mathrm { y } - 100$ , accurately describe this situation?

Accepted Answer

A) True 
 B)False

Question 20

The following could be graphs of solutions to which of the following differential equations?

Accepted Answer

A)  $y ^ { \prime }$ = 3y(y - 2) 
B)  $y ^ { \prime }$ = $y ^ { 2 }$ + 2 
C)  $y ^ { \prime }$ = y(y + 2) 
D)  $y ^ { \prime }$ = (y - 2) $e ^ { y }$ 
E)  none of these 
A)  $y ^ { \prime }$ = 3y(y - 2) 
B)  $y ^ { \prime }$ = $y ^ { 2 }$ + 2 
C)  $y ^ { \prime }$ = y(y + 2) 
D)  $y ^ { \prime }$ = (y - 2) $e ^ { y }$ 
E)  none of these

Let f(t) be the solution to of $y ^ { \prime }$ = $y ^ { 2 }$ t - y, y(0) = 1. Which of the following statements is true?

Given the differential equation with the given initial condition: $y ^ { \prime } = e ^ { - y } ; \quad y ( 0 ) = 0$ is this the solution $y = \ln | t + 1 | ?$

Solve the differential equation with the given initial condition. - $y ^ { \prime } = \frac { x } { y ^ { \prime } } y ( 3 ) = 5$

Find a constant solution of $y ^ { \prime } = 10 y - 7$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Given the differential equation: $\left( t ^ { 2 } + 1 \right) y ^ { \prime } = y t$ , is this the solution $y = c \sqrt { t ^ { 2 } + 1 } ?$

Given the differential equation: y' is this the solution $y = \pm \sqrt { 2 \ln | t | + C } ?$

How much would you need to invest per month - in effect, continuously - in an investment account that pays an annual interest rate of 9%, compounded continuously, in order for the account to be worth $100,000 after 20 years? Enter just an integer representing dollars to the nearest dollar (no units)

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. 2t $y ^ { \prime }$ - y = $\frac { 6 } { t }$ ; y(1) = -1, t > 0

Solve the differential equation with the given initial condition. - $y ^ { \prime } = t y , y ( 0 ) = - 1$

Find f'(1) if f(t) is a solution to the initial value problem: $y ^ { \prime } = t y ^ { 2 } + 5 , y ( 1 ) = 1$ Enter just an integer.

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false: -f(0) = 1

Given the differential equation: $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { \mathrm { t } + \mathrm { e } ^ { \mathrm { t } } } { \mathrm { y } } \text {, }$ is this the solution $y = \pm \sqrt { t ^ { 2 } + 2 e ^ { t } + C } ?$

Solve the differential equation with the given initial condition. - $y ^ { \prime } = y ( t - 2 ) ; y ( 0 ) = 1$

The following could be graphs of solutions to which of the following differential equations?

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

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Exam 10: Differential Equations

Let f(t) be the solution to of y′y ^ { \prime }y′ = y2y ^ { 2 }y2 t - y, y(0) = 1. Which of the following statements is true?

Given the differential equation with the given initial condition: y′=e−y;y(0)=0y ^ { \prime } = e ^ { - y } ; \quad y ( 0 ) = 0y′=e−y;y(0)=0 is this the solution y=ln⁡∣t+1∣?y = \ln | t + 1 | ?y=ln∣t+1∣?

Solve the differential equation with the given initial condition. - y′=xy′y(3)=5y ^ { \prime } = \frac { x } { y ^ { \prime } } y ( 3 ) = 5y′=y′x​y(3)=5

Use Euler's method with n = 4 on the interval 0≤t≤20 \leq t \leq 20≤t≤2 to approximate the solution f(t) to y′=y−4ty ^ { \prime } = y - 4 ty′=y−4t , y(0)=2y ( 0 ) = 2y(0)=2 Is the following graph accurate?

Find a constant solution of y′=10y−7y ^ { \prime } = 10 y - 7y′=10y−7 Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

Given the differential equation: (t2+1)y′=yt\left( t ^ { 2 } + 1 \right) y ^ { \prime } = y t(t2+1)y′=yt , is this the solution y=ct2+1?y = c \sqrt { t ^ { 2 } + 1 } ?y=ct2+1​?

Given the differential equation: y' is this the solution y=±2ln⁡∣t∣+C?y = \pm \sqrt { 2 \ln | t | + C } ?y=±2ln∣t∣+C​?

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. 2t y′y ^ { \prime }y′ - y = 6t\frac { 6 } { t }t6​ ; y(1) = -1, t > 0

Solve the differential equation with the given initial condition. - y′=ty,y(0)=−1y ^ { \prime } = t y , y ( 0 ) = - 1y′=ty,y(0)=−1

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false: - f′f ^ { \prime }f′ (32)\left( \frac { 3 } { 2 } \right)(23​) ≈ 32\frac { 3 } { 2 }23​

Find f'(1) if f(t) is a solution to the initial value problem: y′=ty2+5,y(1)=1y ^ { \prime } = t y ^ { 2 } + 5 , y ( 1 ) = 1y′=ty2+5,y(1)=1 Enter just an integer.

The following is a polygonal path obtained from Euler's method with n = 4 to approximate a solution f(t) of a differential equation. Indicate whether the following statements are true or false: -f(0) = 1

Given the differential equation: dydt=t+ety, \frac { \mathrm { dy } } { \mathrm { dt } } = \frac { \mathrm { t } + \mathrm { e } ^ { \mathrm { t } } } { \mathrm { y } } \text {, }dtdy​=yt+et​, is this the solution y=±t2+2et+C?y = \pm \sqrt { t ^ { 2 } + 2 e ^ { t } + C } ?y=±t2+2et+C​?

Solve the differential equation with the given initial condition. - y′=y(t−2);y(0)=1y ^ { \prime } = y ( t - 2 ) ; y ( 0 ) = 1y′=y(t−2);y(0)=1

The following could be graphs of solutions to which of the following differential equations?

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

Let f(t) be the solution to of $y ^ { \prime }$ = $y ^ { 2 }$ t - y, y(0) = 1. Which of the following statements is true?

Given the differential equation with the given initial condition: $y ^ { \prime } = e ^ { - y } ; \quad y ( 0 ) = 0$ is this the solution $y = \ln | t + 1 | ?$

Solve the differential equation with the given initial condition. - $y ^ { \prime } = \frac { x } { y ^ { \prime } } y ( 3 ) = 5$

Find a constant solution of $y ^ { \prime } = 10 y - 7$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Given the differential equation: $\left( t ^ { 2 } + 1 \right) y ^ { \prime } = y t$ , is this the solution $y = c \sqrt { t ^ { 2 } + 1 } ?$

Given the differential equation: y' is this the solution $y = \pm \sqrt { 2 \ln | t | + C } ?$

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. 2t $y ^ { \prime }$ - y = $\frac { 6 } { t }$ ; y(1) = -1, t > 0

Solve the differential equation with the given initial condition. - $y ^ { \prime } = t y , y ( 0 ) = - 1$

Find f'(1) if f(t) is a solution to the initial value problem: $y ^ { \prime } = t y ^ { 2 } + 5 , y ( 1 ) = 1$ Enter just an integer.

Given the differential equation: $\frac { \mathrm { dy } } { \mathrm { dt } } = \frac { \mathrm { t } + \mathrm { e } ^ { \mathrm { t } } } { \mathrm { y } } \text {, }$ is this the solution $y = \pm \sqrt { t ^ { 2 } + 2 e ^ { t } + C } ?$

Solve the differential equation with the given initial condition. - $y ^ { \prime } = y ( t - 2 ) ; y ( 0 ) = 1$