Question 1

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

Accepted Answer

A)    y′y ^ { \prime }y′  = 3y(y - 2) 
B)    y′y ^ { \prime }y′  =  y2y ^ { 2 }y2  + 2
C)    y′y ^ { \prime }y′  = y(y + 2) 
D)    y′y ^ { \prime }y′  = (y - 2)   eye ^ { y }ey 
E)   none of these F) C) and E)
G) B) and D)

Question 2

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

Accepted Answer

A)   integrating factor:\  t−1/2t ^ { - 1 / 2 }t−1/2 General solution: y = -  2t\frac { 2 } { t }t2​  + C  t1/2t ^ {  1 / 2 }t1/2 Particular solution: y = -  2t\frac { 2 } { t }t2​  +  t1/2t ^ { 1 / 2 }t1/2 
B)   integrating factor: 2tGeneral solution: y = -  34t2\frac { 3 } { 4 t ^ { 2 } }4t23​  +  C2t\frac { C } { 2 t }2tC​ Particular solution: y = -  34t2\frac { 3 } { 4 t ^ { 2 } }4t23​  +  14t\frac { 1 } { 4 t }4t1​ 
C)   integrating factor:  t−1/2t ^ { - 1 / 2 }t−1/2 General solution: y = -4 + C  t1/2t ^ {  1 / 2 }t1/2 Particular solution: y = -4 + 3  t1/2t ^ {  1 / 2 }t1/2 
D)   integrating factor: 2tGeneral solution: y = -  32t2\frac { 3 } { 2 t ^ { 2 } }2t23​  +  C2t\frac { C } { 2 t }2tC​ Particular solution: y = -  32t2\frac { 3 } { 2 t ^ { 2 } }2t23​  +  12t\frac { 1 } { 2 t }2t1​E) B) and C)
F) C) and D)

Question 3

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

Accepted Answer

A) True 
 B)False

Question 5

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 6

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 7

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 8

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

Accepted Answer

A) True 
 B)False

Question 9

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?

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′f ^ { \prime }f′  (1)  = 0 E) B) and C)
F) A) and D)

Question 10

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 11

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

Accepted Answer

A)   y =  et+1e ^ { t + 1 }et+1 
B)   y = -  et2/2\mathrm { e } t^ { 2 } / 2et2/2 
C)   y =  t22\frac { t ^ { 2 } } { 2 }2t2​ 
D)   y = -1 +  et2/2\mathrm { e } t^ { 2 } / 2et2/2E) None of the above
F) A) and C)

Question 12

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 13

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 14

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 15

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 16

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 17

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 18

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.
-Let  y′y ^ { \prime }y′  = 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 F) C) and D)
G) A) and E)

Question 19

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

Accepted Answer

A)   y =  x2+9\sqrt { x ^ { 2 } + 9 }x2+9c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
B)   y = ±  x2+16\sqrt { x ^ { 2 } + 16 }x2+16c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
C)   y = ±  x2+9\sqrt { x ^ { 2 } + 9 }x2+9c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​ 
D)   y =  x2+16\sqrt { x ^ { 2 } + 16 }x2+16c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​E) A) and B)
F) A) and D)

Question 20

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

Accepted Answer

A)   y =  e[(t2/2) −2t]e \left[ \left( t ^ { 2 } / 2 \right)  - 2 t \right]e[(t2/2) −2t]  - 2t + 1
B)   y =  t22\frac { t ^ { 2 } } { 2 }2t2​  - 2t
C)   y =  e[(t2/2) −2t]e \left[ \left( t ^ { 2 } / 2 \right)  - 2 t \right]e[(t2/2) −2t] 
D)   y = 0
E)   none of these F) C) and D)
G) B) and E)

Exam 10: Differential Equations

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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.

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

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)

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

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?

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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∣?

Correct AnswerverifiedShow Answer

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.

Correct AnswerverifiedShow Answer

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​

Correct AnswerverifiedShow Answer

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​?

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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​?

Correct AnswerverifiedShow Answer

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​ .

Correct AnswerverifiedShow Answer

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

Correct AnswerverifiedShow Answer

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?

Correct AnswerverifiedShow Answer

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

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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.

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

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

Correct Answer
verified

Correct Answer
verified