Question 1

Which of the following is a sketch of the solution of $y ^ { \prime }$ = $y ^ { 2 }$ - 9; y(0) = 2 ?

Accepted Answer

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

Question 2

Let f(t) be the solution of $y ^ { \prime }$ = $y ^ { 2 }$ t + y + $\mathrm { e } ^ { \mathrm { t } }$ , f(0) = 2. If Euler's method with n = 4 is used to approximate f(t) for $0 \leq t \leq 2$ find f $\left( \frac { 1 } { 2 } \right)$ .

Accepted Answer

A)  2(4 + $\sqrt { \mathrm { e } }$ ) 
B)  $\frac { y ^ { 2 } } { 2 }$ + y + $e ^ { 1 / 2 }$ 
C)  $\frac { 7 } { 2 }$ 
D)  3 + $e ^ { 2 }$ 
E)  none of these 
A)  2(4 + $\sqrt { \mathrm { e } }$ ) 
B)  $\frac { y ^ { 2 } } { 2 }$ + y + $e ^ { 1 / 2 }$ 
C)  $\frac { 7 } { 2 }$ 
D)  3 + $e ^ { 2 }$ 
E)  none of these

Question 3

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

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. $y ^ { \prime } = y ^ { 2 } - 2 y - 8 ; y ( 0 ) = - 3 ; y ( 0 ) = 3$ Do these graphs represent the situation?

Accepted Answer

A) True 
 B)False

Question 5

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 2, then the corresponding solution has an inflection point.

Accepted Answer

A) True 
 B)False

Question 6

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 = -3, y = 1, and y = 5 are the constant solutions to $y ^ { \prime }$ = g(y).

Accepted Answer

A) True 
 B)False

Question 7

Given the differential equation: $ty ^ { \prime } = \ln t$ , is this the solution $y = \frac { ( \ln t ) ^ { 2 } } { 2 } + C ?$

Accepted Answer

A) True 
 B)False

Question 8

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

Accepted Answer

A)  y = ln $| \tan \mathrm { t } |$ + 1 
B)  y = $\frac { \tan ^ { 2 } \mathrm { t } } { 2 }$ + 1 
C)  y = tan t + 1 
D)  y = $\frac { \sec ^ { 2 } t } { 3 }$ + $\frac { 2 } { 3 }$ 
A)  y = ln $| \tan \mathrm { t } |$ + 1 
B)  y = $\frac { \tan ^ { 2 } \mathrm { t } } { 2 }$ + 1 
C)  y = tan t + 1 
D)  y = $\frac { \sec ^ { 2 } t } { 3 }$ + $\frac { 2 } { 3 }$

Question 9

A savings account earns 6% annual interest, compounded continuously. An initial deposit of $8500 is made, and thereafter money is withdrawn continuously at the rate of $480 per year. Does the following accurately represent this situation: $y ^ { \prime } = 0.06 y - 480 ; y ( 0 ) = 8500 ?$

Accepted Answer

A) True 
 B)False

Question 10

v
-An initial deposit of $8,000 is made into an account earning 6.5% compounded continuously. Thereafter, money is deposited into the account at a constant rate of $2600 per year. Find the amount in this account at any time t. How much is in this account after 5 years?

Accepted Answer

A)  A = 60,000 $e ^ { 0.065 t }$ - 52,000= $31,041.84 
B)  A = 52,000 $e ^ { 0.065 t }$ - 44,000= $27,969.59 
C)  A = 48,000 $e ^ { 0.065 t }$ - 40,000= $26,433.47 
D)  A = 44,000 $e ^ { 0.065 t }$ - 36,000= $24,897.34 
A)  A = 60,000 $e ^ { 0.065 t }$ - 52,000= $31,041.84 
B)  A = 52,000 $e ^ { 0.065 t }$ - 44,000= $27,969.59 
C)  A = 48,000 $e ^ { 0.065 t }$ - 40,000= $26,433.47 
D)  A = 44,000 $e ^ { 0.065 t }$ - 36,000= $24,897.34

Question 11

Consider the differential equation y' = y - $y ^ { 2 }$ . Which of the following statements is/are true?

Accepted Answer

A)  The function f(t) = $\frac { 1 } { \left( 1 + e ^ { - t } \right) }$ is a solution to this differential equation with initial condition $y ( 0 ) = \frac { 1 } { 2 }$ 
B)  This differential equation has infinitely many solutions. 
C)  The constant function f(t) = 1 is a solution to this differential equation. 
D)  If f(t) is a solution to the differential equation satisfying the initial condition y(0) = 0, then $f ^ { \prime } ( 0 ) = 0$ 
E)  All of these statements are true. 
A)  The function f(t) = $\frac { 1 } { \left( 1 + e ^ { - t } \right) }$ is a solution to this differential equation with initial condition $y ( 0 ) = \frac { 1 } { 2 }$ 
B)  This differential equation has infinitely many solutions. 
C)  The constant function f(t) = 1 is a solution to this differential equation. 
D)  If f(t) is a solution to the differential equation satisfying the initial condition y(0) = 0, then $f ^ { \prime } ( 0 ) = 0$ 
E)  All of these statements are true.

Question 12

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. $y ^ { \prime }$ - 4y = -2 $e ^ { 2 t }$ ; y(0) = -1

Accepted Answer

A)  integrating factor: $e ^ { - 4 t }$
General solution: $y = - e ^ { - 2 t } + C ^ { e ^ { 4 t } }$
Particular solution: y = - $e ^ { - 2 t }$ - 2 $e ^ { 4 t }$ 
B)  integrating factor: $e ^ { - 4 t }$
General solution: y = $e ^ { 2 t }$ + C $e ^ { 4 t }$
Particular solution: y = $e ^ { 2 t }$ - 2 $e ^ { 4 t }$ 
C)  integrating factor: $e ^ { 4 t }$
General solution: y = -2t + C $e ^ { 4 t }$
Particular solution: y = -2t - $e ^ { 4 t }$ 
D)  integrating factor: $e ^ { 4 t }$
General solution: y = $\frac { 1 } { 3 }$ $e ^ { 6 t }$ + C $e ^ { 4 t }$
Particular solution: y = $\frac { 1 } { 3 }$ $e ^ { 6 t }$ - $\frac { 4 } { 3 }$ $e ^ { 4 t }$
Solve the equation using an integrating factor. 
A)  integrating factor: $e ^ { - 4 t }$
General solution: $y = - e ^ { - 2 t } + C ^ { e ^ { 4 t } }$
Particular solution: y = - $e ^ { - 2 t }$ - 2 $e ^ { 4 t }$ 
B)  integrating factor: $e ^ { - 4 t }$
General solution: y = $e ^ { 2 t }$ + C $e ^ { 4 t }$
Particular solution: y = $e ^ { 2 t }$ - 2 $e ^ { 4 t }$ 
C)  integrating factor: $e ^ { 4 t }$
General solution: y = -2t + C $e ^ { 4 t }$
Particular solution: y = -2t - $e ^ { 4 t }$ 
D)  integrating factor: $e ^ { 4 t }$
General solution: y = $\frac { 1 } { 3 }$ $e ^ { 6 t }$ + C $e ^ { 4 t }$
Particular solution: y = $\frac { 1 } { 3 }$ $e ^ { 6 t }$ - $\frac { 4 } { 3 }$ $e ^ { 4 t }$
Solve the equation using an integrating factor.

Question 13

Use Euler's method with n = 2 to approximate the solution f(t) to $y ^ { \prime } = 2 y - t , y ( 0 ) = 1$ Estimate f(1).
Enter just a reduced fraction of form $\frac { a } { b }$ .

Accepted Answer

The answer of Use Euler's method with n = 2...

Question 14

A certain drug is introduced into a person's bloodstream. Suppose that the rate of decrease of the concentration of the drug in the blood is directly proportional to the product of two quantities: (a) the amount of time elapsed since the drug was introduced, and (b) the square of the concentration. Let y = f(t) denote the concentration of the drug in the blood at time t. Set up a differential equation satisfied by f(t). Does the following accurately describe this situation: $y ^ { \prime } = k t y ^ { 2 } , \text { where } k \text { is a negative constant }$

Accepted Answer

A) True 
 B)False

Question 15

Combine the terms y and $y ^ { \prime }$ into the derivative of a product: $y ^ { \prime } \tan t + y \sec ^ { 2 } t = 1$ .
Is this derivative correct: $\frac { \mathrm { d } } { \mathrm { dt } } [ \mathrm { y } \tan \mathrm { t } ] = 1$ ?

Accepted Answer

A) True 
 B)False

Question 16

Combine the terms y and $y ^ { \prime }$ into the derivative of a product, then solve the equation. $\mathrm { e } ^ { 3 \mathrm { t } ^ { 2 } } \mathrm { y } ^ { \prime } + 6 \mathrm { te } ^ { 3 \mathrm { t } ^ { 2 } } \mathrm { y } = \frac { 5 \sqrt { \mathrm { t } } } { 4 }$. Is this the solution: $y = \frac { 5 } { 6 } t ^ { 3 / 2 } e ^ { - 3 t ^ { 2 } } + C e ^ { - 3 t ^ { 2 } } ?$

Accepted Answer

A) True 
 B)False

Question 17

Suppose that a substance A is converted to substance B at a rate that is proportional to the cube of the amount of B present. The amount of A and B together is always constant, say M. If f( t) = y is the amount of A present at time t, then which of the following differential equation describes the situation?

Accepted Answer

A) $y ^ { \prime }$ = $k ( M - y ) ^ { 3 }$ ; k 0 C) $y ^ { \prime }$ = $\mathrm { ky } ^ { 3 }$ ; k > 0 D) $y ^ { \prime }$ = $\mathrm { ky } ^ { 3 }$ ; k < 0 E) none of these A) $y ^ { \prime }$ = $k ( M - y ) ^ { 3 }$ ; k 0 C) $y ^ { \prime }$ = $\mathrm { ky } ^ { 3 }$ ; k > 0 D) $y ^ { \prime }$ = $\mathrm { ky } ^ { 3 }$ ; k < 0 E) none of these

Question 18

Solve the initial value problem using an integrating factor. 
-t $y ^ { \prime }$ + 3y = 5t; $y ( 1 ) = 1$ , t > 0

Accepted Answer

A)  y = $\frac { 5 } { 4 }$ t - $\frac { 1 } { 4 t ^ { 3 } }$ 
B)  y = $\sqrt { \frac { 4 } { 3 } t ^ { 2 } }$ + 1 - $\sqrt { \frac { 4 } { 3 } }$ 
C)  y = 5 $t ^ { 2 }$ - 4t 
D)  y = 5 $t ^ { 3 }$ - 4 $t ^ { 2 }$ 
A)  y = $\frac { 5 } { 4 }$ t - $\frac { 1 } { 4 t ^ { 3 } }$ 
B)  y = $\sqrt { \frac { 4 } { 3 } t ^ { 2 } }$ + 1 - $\sqrt { \frac { 4 } { 3 } }$ 
C)  y = 5 $t ^ { 2 }$ - 4t 
D)  y = 5 $t ^ { 3 }$ - 4 $t ^ { 2 }$

Question 19

Solve the differential equation with the given initial condition. 
-$y ^ { \prime } = 3 t ^ { 2 } ( 4 - y ) ^ { 2 } , y ( 0 ) = 2$

Accepted Answer

A)  y = 4 - $\frac { 1 } { t ^ { 3 } + \frac { 1 } { 2 } }$ 
B)  y = $\frac { 1 } { t ^ { 3 } + 2 }$ 
C)  y = $t ^ { 3 }$ + $\frac { 1 } { 2 }$ 
D)  y = 4 + $t ^ { 3 }$ 
A)  y = 4 - $\frac { 1 } { t ^ { 3 } + \frac { 1 } { 2 } }$ 
B)  y = $\frac { 1 } { t ^ { 3 } + 2 }$ 
C)  y = $t ^ { 3 }$ + $\frac { 1 } { 2 }$ 
D)  y = 4 + $t ^ { 3 }$

Question 20

Which of the following functions solves the differential equation: $y ^ { \prime } = e ^ { - 2 x } + 3 ?$

Accepted Answer

A)  y = $e ^ { - 2 x }$ + 3 
B)  y = $\frac { 1 } { 2 }$ $e ^ { - 2 x }$ + 3 
C)  y = - $\frac { 1 } { 2 }$ $e ^ { - 2 x }$ + 3x 
D)  none of these 
A)  y = $e ^ { - 2 x }$ + 3 
B)  y = $\frac { 1 } { 2 }$ $e ^ { - 2 x }$ + 3 
C)  y = - $\frac { 1 } { 2 }$ $e ^ { - 2 x }$ + 3x 
D)  none of these

Which of the following is a sketch of the solution of $y ^ { \prime }$ = $y ^ { 2 }$ - 9; y(0) = 2 ?

Let f(t) be the solution of $y ^ { \prime }$ = $y ^ { 2 }$ t + y + $\mathrm { e } ^ { \mathrm { t } }$ , f(0) = 2. If Euler's method with n = 4 is used to approximate f(t) for $0 \leq t \leq 2$ find f $\left( \frac { 1 } { 2 } \right)$ .

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

Given the differential equation: $ty ^ { \prime } = \ln t$ , is this the solution $y = \frac { ( \ln t ) ^ { 2 } } { 2 } + C ?$

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

A savings account earns 6% annual interest, compounded continuously. An initial deposit of $8500 is made, and thereafter money is withdrawn continuously at the rate of $480 per year. Does the following accurately represent this situation: $y ^ { \prime } = 0.06 y - 480 ; y ( 0 ) = 8500 ?$

v -An initial deposit of $8,000 is made into an account earning 6.5% compounded continuously. Thereafter, money is deposited into the account at a constant rate of $2600 per year. Find the amount in this account at any time t. How much is in this account after 5 years?

Consider the differential equation y' = y - $y ^ { 2 }$ . Which of the following statements is/are true?

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. $y ^ { \prime }$ - 4y = -2 $e ^ { 2 t }$ ; y(0) = -1

Use Euler's method with n = 2 to approximate the solution f(t) to $y ^ { \prime } = 2 y - t , y ( 0 ) = 1$ Estimate f(1). Enter just a reduced fraction of form $\frac { a } { b }$ .

Combine the terms y and $y ^ { \prime }$ into the derivative of a product: $y ^ { \prime } \tan t + y \sec ^ { 2 } t = 1$ . Is this derivative correct: $\frac { \mathrm { d } } { \mathrm { dt } } [ \mathrm { y } \tan \mathrm { t } ] = 1$ ?

Solve the initial value problem using an integrating factor. -t $y ^ { \prime }$ + 3y = 5t; $y ( 1 ) = 1$ , t > 0

Solve the differential equation with the given initial condition. - $y ^ { \prime } = 3 t ^ { 2 } ( 4 - y ) ^ { 2 } , y ( 0 ) = 2$

Which of the following functions solves the differential equation: $y ^ { \prime } = e ^ { - 2 x } + 3 ?$

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

Which of the following is a sketch of the solution of y′y ^ { \prime }y′ = y2y ^ { 2 }y2 - 9; y(0) = 2 ?

Let f(t) be the solution of y′y ^ { \prime }y′ = y2y ^ { 2 }y2 t + y + et\mathrm { e } ^ { \mathrm { t } }et , f(0) = 2. If Euler's method with n = 4 is used to approximate f(t) for 0≤t≤20 \leq t \leq 20≤t≤2 find f (12)\left( \frac { 1 } { 2 } \right)(21​) .

Given the differential equation with the given initial condition: y′=tcos⁡t;y(0)=0\mathrm { y } ^ { \prime } = \mathrm { t } \cos \mathrm { t } ; \mathrm { y } ( 0 ) = 0y′=tcost;y(0)=0 is this the solution y=tsin⁡t+cos⁡t−1?y = t \sin t + \cos t - 1 ?y=tsint+cost−1?

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 2, then the corresponding solution has an inflection point.

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 = -3, y = 1, and y = 5 are the constant solutions to y′y ^ { \prime }y′ = g(y).

Given the differential equation: ty′=ln⁡tty ^ { \prime } = \ln tty′=lnt , is this the solution y=(ln⁡t)22+C?y = \frac { ( \ln t ) ^ { 2 } } { 2 } + C ?y=2(lnt)2​+C?

Solve the differential equation with the given initial condition. - y′=tan⁡tsec⁡2t;y(0)=1y ^ { \prime } = \tan t \sec ^ { 2 } t ; y ( 0 ) = 1y′=tantsec2t;y(0)=1

v -An initial deposit of $8,000 is made into an account earning 6.5% compounded continuously. Thereafter, money is deposited into the account at a constant rate of $2600 per year. Find the amount in this account at any time t. How much is in this account after 5 years?

Consider the differential equation y' = y - y2y ^ { 2 }y2 . Which of the following statements is/are true?

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. y′y ^ { \prime }y′ - 4y = -2 e2te ^ { 2 t }e2t ; y(0) = -1

Use Euler's method with n = 2 to approximate the solution f(t) to y′=2y−t,y(0)=1y ^ { \prime } = 2 y - t , y ( 0 ) = 1y′=2y−t,y(0)=1 Estimate f(1). Enter just a reduced fraction of form ab\frac { a } { b }ba​ .

Solve the initial value problem using an integrating factor. -t y′y ^ { \prime }y′ + 3y = 5t; y(1)=1y ( 1 ) = 1y(1)=1 , t > 0

Solve the differential equation with the given initial condition. - y′=3t2(4−y)2,y(0)=2y ^ { \prime } = 3 t ^ { 2 } ( 4 - y ) ^ { 2 } , y ( 0 ) = 2y′=3t2(4−y)2,y(0)=2

Which of the following functions solves the differential equation: y′=e−2x+3?y ^ { \prime } = e ^ { - 2 x } + 3 ?y′=e−2x+3?

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

Which of the following is a sketch of the solution of $y ^ { \prime }$ = $y ^ { 2 }$ - 9; y(0) = 2 ?

Let f(t) be the solution of $y ^ { \prime }$ = $y ^ { 2 }$ t + y + $\mathrm { e } ^ { \mathrm { t } }$ , f(0) = 2. If Euler's method with n = 4 is used to approximate f(t) for $0 \leq t \leq 2$ find f $\left( \frac { 1 } { 2 } \right)$ .

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

Given the differential equation: $ty ^ { \prime } = \ln t$ , is this the solution $y = \frac { ( \ln t ) ^ { 2 } } { 2 } + C ?$

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

Consider the differential equation y' = y - $y ^ { 2 }$ . Which of the following statements is/are true?

Find the integrating factor, the general solution, and the particular solution satisfying the initial condition. $y ^ { \prime }$ - 4y = -2 $e ^ { 2 t }$ ; y(0) = -1

Use Euler's method with n = 2 to approximate the solution f(t) to $y ^ { \prime } = 2 y - t , y ( 0 ) = 1$ Estimate f(1). Enter just a reduced fraction of form $\frac { a } { b }$ .

Solve the initial value problem using an integrating factor. -t $y ^ { \prime }$ + 3y = 5t; $y ( 1 ) = 1$ , t > 0

Solve the differential equation with the given initial condition. - $y ^ { \prime } = 3 t ^ { 2 } ( 4 - y ) ^ { 2 } , y ( 0 ) = 2$

Which of the following functions solves the differential equation: $y ^ { \prime } = e ^ { - 2 x } + 3 ?$