Question 1

Find the solution of the differential equation that satisfies the initial condition $y ( 0 ) = 1$.
$$\frac { d y } { d x } = 6 x ^ { 5 } y$$

Accepted Answer

$$y = e ^ { x ^ { 6 } }$$

Question 2

An object with mass $m$ is dropped from rest and we assume that the air resistance is proportional to the speed of the object. If $s ( t )$ is the distance dropped after $t$ seconds, then the speed is $v = s ^ { t } ( t )$ and the acceleration is $a = v ^ { \prime } ( t )$. If $g$ is the acceleration due to gravity, then the downward force on the object is $m g - c v$, where $c$ is a positive constant, and Newton's Second Law gives $m \frac { d v } { d t } = m g - c v$.
Find the limiting velocity.

Accepted Answer

$$\nu = \frac { m g } { c }$$

Question 3

A sum of $\$ 5,000$ is invested at $20 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$.

Accepted Answer

$$\frac { d A } { d t } = 0.20 A , \quad A ( 0 ) = 5,000$$

Question 4

Solve the initial-value problem.
$$\frac { d r } { d t } + 2 t r - r = 0 , \quad r ( 0 ) = 10$$

Accepted Answer

The answer of Solve the initial-value problem.
\[\frac { d r...

Question 5

Use Euler's method with step size $0.25$ to estimate $y ( 1 )$, where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places.
$$y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0$$

Accepted Answer

A)  $y ( 1 ) = 4.1622$ 
B)  $y ( 1 ) = 1.6622$ 
C)  $y ( 1 ) = 3.1622$ 
D)  $y ( 1 ) = 2.6622$ 
A)  $y ( 1 ) = 4.1622$ 
B)  $y ( 1 ) = 1.6622$ 
C)  $y ( 1 ) = 3.1622$ 
D)  $y ( 1 ) = 2.6622$

Question 6

Let $c$ be a positive number. A differential equation of the form
$$\frac { d y } { d t } = k y ^ { 1 + c }$$ where $k$ is a positive constant is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 1.04 }$. If 5 such rabbits breed initially and the warren has 23 rabbits after 5 months, then when is doomsday? Select the correct answer.

Accepted Answer

A)  $138.06$ months 
B)  $158.06$ months 
C)  $118.06$ months 
D)  $88.06$ months 
E)  $68.06$ months 
A)  $138.06$ months 
B)  $158.06$ months 
C)  $118.06$ months 
D)  $88.06$ months 
E)  $68.06$ months

Question 7

Determine whether the differential equation is linear. Select the correct answer.
$$y ^ { t } + 6 x ^ { 5 } y = 6 x ^ { 5 }$$

Accepted Answer

A)  the equation is linear 
B)  the equation is not linear 
A)  the equation is linear 
B)  the equation is not linear

Question 8

Let $c$ be a positive number. A differential equation of the form
$$\frac { d y } { d t } = k y ^ { 1 + c }$$
where $k$ is a positive constant is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 106 }$. If 3 such rabbits breed initially and the warren has 27 rabbits after 3 months, then when is doomsday?

Accepted Answer

A)  $47.03$ months 
B)  $97.03$ months 
C)  $67.03$ months 
D)  $137.03$ months 
E)  $117.03$ months 
A)  $47.03$ months 
B)  $97.03$ months 
C)  $67.03$ months 
D)  $137.03$ months 
E)  $117.03$ months

Question 9

Solve the initial-value problem.
$$x y ^ { t } - y = x \ln x , y ( 1 ) = 3$$

Accepted Answer

The answer of Solve the initial-value problem.
\[x y ^ {...

Question 10

Solve the differential equation.
$3 y y ^ { \prime } = 7 x$

Accepted Answer

A)  $7 x ^ { 2 } + 3 y ^ { 2 } = C$ 
B)  $3 x ^ { 2 } + 7 y ^ { 2 } = C$ 
C)  $3 x ^ { 2 } - 7 y ^ { 2 } = C$ 
D)  $7 x ^ { 2 } - 3 y ^ { 2 } = C$ 
E)  $7 x ^ { 2 } + 3 y ^ { 2 } = 10$ 
A)  $7 x ^ { 2 } + 3 y ^ { 2 } = C$ 
B)  $3 x ^ { 2 } + 7 y ^ { 2 } = C$ 
C)  $3 x ^ { 2 } - 7 y ^ { 2 } = C$ 
D)  $7 x ^ { 2 } - 3 y ^ { 2 } = C$ 
E)  $7 x ^ { 2 } + 3 y ^ { 2 } = 10$

Question 11

For what nonzero values of $k$ does the function $y = A \sin k t + B \cos k t$ satisfy the differential equation $y ^ { \prime \prime } + 100 y = 0$ for all values of $A$ and $B$ ?

Accepted Answer

A)  $k = 10$ 
B)  $k = - 100$ 
C)  $k = - 10$ 
D)  $k = 100$ 
E)  $k = 1$ 
A)  $k = 10$ 
B)  $k = - 100$ 
C)  $k = - 10$ 
D)  $k = 100$ 
E)  $k = 1$

Question 12

Solve the initial-value problem. $x y ^ { t} = y + x ^ { 2 } \sin x , y ( 11 \pi ) = 0$

Accepted Answer

A)  $y = 11 x \sin x$ 
B)  $y = - x \sin x + 11 x$ 
C)  $y = - x \cos x - x$ 
D)  $y = 11 x \cos x - \sin x$ 
E)  $y = x \sin x - 11 x$ 
A)  $y = 11 x \sin x$ 
B)  $y = - x \sin x + 11 x$ 
C)  $y = - x \cos x - x$ 
D)  $y = 11 x \cos x - \sin x$ 
E)  $y = x \sin x - 11 x$

Question 13

A sum of $\$ 2,000$ is invested at $20 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$.

Accepted Answer

A)  $\frac { d A } { d t } = 20 A , \quad A ( 0 ) = 3,000$ 
B)  $\frac { d A } { d t } = 1,500 A , \quad A ( 0 ) = 2,000$ 
C)  $\frac { d A } { d t } = 0.20 A , A ( 0 ) = 200$ 
D)  $\frac { d A } { d t } = 0.20 A ( 0 ) , \quad A ( 0 ) = 2,000$ 
E)  $\frac { d A } { d t } = 0.20 A , \quad A ( 0 ) = 2,000$ 
A)  $\frac { d A } { d t } = 20 A , \quad A ( 0 ) = 3,000$ 
B)  $\frac { d A } { d t } = 1,500 A , \quad A ( 0 ) = 2,000$ 
C)  $\frac { d A } { d t } = 0.20 A , A ( 0 ) = 200$ 
D)  $\frac { d A } { d t } = 0.20 A ( 0 ) , \quad A ( 0 ) = 2,000$ 
E)  $\frac { d A } { d t } = 0.20 A , \quad A ( 0 ) = 2,000$

Question 14

Find the orthogonal trajectories of the family of curves.
$$y = k x ^ { 9 }$$

Accepted Answer

A)  $x ^ { 2 } - 8 y ^ { 2 } = 0$ 
B)  $x ^ { 2 } - 8 y ^ { 2 } = C$ 
C)  $x ^ { 2 } - y ^ { 2 } = 0$ 
D)  $x ^ { 2 } + 8 y ^ { 2 } = 0$ 
E)  $x ^ { 2 } + 9 y ^ { 2 } = C$ 
A)  $x ^ { 2 } - 8 y ^ { 2 } = 0$ 
B)  $x ^ { 2 } - 8 y ^ { 2 } = C$ 
C)  $x ^ { 2 } - y ^ { 2 } = 0$ 
D)  $x ^ { 2 } + 8 y ^ { 2 } = 0$ 
E)  $x ^ { 2 } + 9 y ^ { 2 } = C$

Question 15

Solve the differential equation.
$$4 \frac { d w } { d t } + 7 e ^ { t + w } = 0$$

Accepted Answer

The answer of Solve the differential equation.
\[4 \frac { d...

Question 16

$y = C e ^ { 4 x ^ { 2 } }$ is the solution of the differential equation $y ^ { \prime } = 8 x y$. Find the solution that satisfies the initial condition $y ( 1 ) = 1$.

Accepted Answer

The answer of \(y = C e ^ { 4...

Question 17

Solve the differential equation. Select the correct answer.
$$y ^ { t } = x ^ { 2 } e ^ { - \sin x } - y \cos x$$

Accepted Answer

A)  $y = 2 \mathrm { Ce } e ^ { - \sin x }$ 
B)  $y = e ^ { - \sin x } + 2 C x ^ { 2 }$ 
C) 
$y = \frac { 1 } { 3 } x + C e ^ { - \cos x }$ 
D)  $y = \left( 3 x ^ { 3 } + C \right) e ^ { - x } \sin x$ 
E) 
$y = \left( \frac { 1 } { 3 } x ^ { 3 } + C \right) e ^ { - \sin x }$ 
A)  $y = 2 \mathrm { Ce } e ^ { - \sin x }$ 
B)  $y = e ^ { - \sin x } + 2 C x ^ { 2 }$ 
C) 
$y = \frac { 1 } { 3 } x + C e ^ { - \cos x }$ 
D)  $y = \left( 3 x ^ { 3 } + C \right) e ^ { - x } \sin x$ 
E) 
$y = \left( \frac { 1 } { 3 } x ^ { 3 } + C \right) e ^ { - \sin x }$

Question 18

Consider a population $P = P ( t )$ with constant relative birth and death rates $a$ and $\beta$, respectively, and a constant emigration rate $m$, where $\alpha = 0.6 , \beta = 0.9$ and $m = 0.7$. Then the rate of change of the population at time $t$ is modeled by the differential equation
$$\frac { d P } { d t } = k P - m$$
where
$$k = \alpha - \beta$$
Find the solution of this equation with the rate of change of the population at time $t = 3$ that satisfies the initial condition $P ( 0 ) = 2600$.

Accepted Answer

The answer of Consider a population \(P = P (...

Question 19

Determine whether the differential equation is linear.
$$y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }$$

Accepted Answer

the equati

Question 20

Which of the following functions is a solution of the differential equation?
$$y ^ { tt } + 16 y ^ { t } + 64 y = 0$$

Accepted Answer

A)  $y = e ^ { t }$ 
B)  $y = t e ^ { - 3 t }$ 
C)  $y = 6 e ^ { - 8 t }$ 
D)  $y = e ^ { - 8 t }$ 
E)  $y = t ^ { 2 } e ^ { - 8 t }$ 
A)  $y = e ^ { t }$ 
B)  $y = t e ^ { - 3 t }$ 
C)  $y = 6 e ^ { - 8 t }$ 
D)  $y = e ^ { - 8 t }$ 
E)  $y = t ^ { 2 } e ^ { - 8 t }$

Find the solution of the differential equation that satisfies the initial condition $y ( 0 ) = 1$ . $\frac { d y } { d x } = 6 x ^ { 5 } y$

A sum of $\$ 5,000$ is invested at $20 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , \quad r ( 0 ) = 10$

Use Euler's method with step size $0.25$ to estimate $y ( 1 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0$

Determine whether the differential equation is linear. Select the correct answer. $y ^ { t } + 6 x ^ { 5 } y = 6 x ^ { 5 }$

Solve the initial-value problem. $x y ^ { t } - y = x \ln x , y ( 1 ) = 3$

Solve the differential equation. $3 y y ^ { \prime } = 7 x$

For what nonzero values of $k$ does the function $y = A \sin k t + B \cos k t$ satisfy the differential equation $y ^ { \prime \prime } + 100 y = 0$ for all values of $A$ and $B$ ?

Solve the initial-value problem. $x y ^ { t} = y + x ^ { 2 } \sin x , y ( 11 \pi ) = 0$

A sum of $\$ 2,000$ is invested at $20 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Find the orthogonal trajectories of the family of curves. $y = k x ^ { 9 }$

Solve the differential equation. $4 \frac { d w } { d t } + 7 e ^ { t + w } = 0$

$y = C e ^ { 4 x ^ { 2 } }$ is the solution of the differential equation $y ^ { \prime } = 8 x y$ . Find the solution that satisfies the initial condition $y ( 1 ) = 1$ .

Solve the differential equation. Select the correct answer. $y ^ { t } = x ^ { 2 } e ^ { - \sin x } - y \cos x$

Determine whether the differential equation is linear. $y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }$

Which of the following functions is a solution of the differential equation? $y ^ { tt } + 16 y ^ { t } + 64 y = 0$

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Exam 9: Differential Equations

Find the solution of the differential equation that satisfies the initial condition y(0)=1y ( 0 ) = 1y(0)=1 . dydx=6x5y\frac { d y } { d x } = 6 x ^ { 5 } ydxdy​=6x5y

A sum of $5,000\$ 5,000$5,000 is invested at 20%20 \%20% interest. If A(t)A ( t )A(t) is the amount of the investment at time ttt for the case of continuous compounding, write a differential equation and an initial condition satisfied by A(t)A ( t )A(t) .

Solve the initial-value problem. drdt+2tr−r=0,r(0)=10\frac { d r } { d t } + 2 t r - r = 0 , \quad r ( 0 ) = 10dtdr​+2tr−r=0,r(0)=10

Use Euler's method with step size 0.250.250.25 to estimate y(1)y ( 1 )y(1) , where y(x)y ( x )y(x) is the solution of the initial-value problem. Round your answer to four decimal places. yt=4x+y2,y(0)=0y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0yt=4x+y2,y(0)=0

Determine whether the differential equation is linear. Select the correct answer. yt+6x5y=6x5y ^ { t } + 6 x ^ { 5 } y = 6 x ^ { 5 }yt+6x5y=6x5

Solve the initial-value problem. xyt−y=xln⁡x,y(1)=3x y ^ { t } - y = x \ln x , y ( 1 ) = 3xyt−y=xlnx,y(1)=3

Solve the differential equation. 3yy′=7x3 y y ^ { \prime } = 7 x3yy′=7x

For what nonzero values of kkk does the function y=Asin⁡kt+Bcos⁡kty = A \sin k t + B \cos k ty=Asinkt+Bcoskt satisfy the differential equation y′′+100y=0y ^ { \prime \prime } + 100 y = 0y′′+100y=0 for all values of AAA and BBB ?

Solve the initial-value problem. xyt=y+x2sin⁡x,y(11π)=0x y ^ { t} = y + x ^ { 2 } \sin x , y ( 11 \pi ) = 0xyt=y+x2sinx,y(11π)=0

A sum of $2,000\$ 2,000$2,000 is invested at 20%20 \%20% interest. If A(t)A ( t )A(t) is the amount of the investment at time ttt for the case of continuous compounding, write a differential equation and an initial condition satisfied by A(t)A ( t )A(t) .

Find the orthogonal trajectories of the family of curves. y=kx9y = k x ^ { 9 }y=kx9

Solve the differential equation. 4dwdt+7et+w=04 \frac { d w } { d t } + 7 e ^ { t + w } = 04dtdw​+7et+w=0

y=Ce4x2y = C e ^ { 4 x ^ { 2 } }y=Ce4x2 is the solution of the differential equation y′=8xyy ^ { \prime } = 8 x yy′=8xy . Find the solution that satisfies the initial condition y(1)=1y ( 1 ) = 1y(1)=1 .

Solve the differential equation. Select the correct answer. yt=x2e−sin⁡x−ycos⁡xy ^ { t } = x ^ { 2 } e ^ { - \sin x } - y \cos xyt=x2e−sinx−ycosx

Determine whether the differential equation is linear. yt+3x2y=6x2y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }yt+3x2y=6x2

Which of the following functions is a solution of the differential equation? ytt+16yt+64y=0y ^ { tt } + 16 y ^ { t } + 64 y = 0ytt+16yt+64y=0

Functions and Models

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Find the solution of the differential equation that satisfies the initial condition $y ( 0 ) = 1$ . $\frac { d y } { d x } = 6 x ^ { 5 } y$

A sum of $\$ 5,000$ is invested at $20 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Solve the initial-value problem. $\frac { d r } { d t } + 2 t r - r = 0 , \quad r ( 0 ) = 10$

Use Euler's method with step size $0.25$ to estimate $y ( 1 )$ , where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places. $y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0$

Determine whether the differential equation is linear. Select the correct answer. $y ^ { t } + 6 x ^ { 5 } y = 6 x ^ { 5 }$

Solve the initial-value problem. $x y ^ { t } - y = x \ln x , y ( 1 ) = 3$

Solve the differential equation. $3 y y ^ { \prime } = 7 x$

For what nonzero values of $k$ does the function $y = A \sin k t + B \cos k t$ satisfy the differential equation $y ^ { \prime \prime } + 100 y = 0$ for all values of $A$ and $B$ ?

Solve the initial-value problem. $x y ^ { t} = y + x ^ { 2 } \sin x , y ( 11 \pi ) = 0$

A sum of $\$ 2,000$ is invested at $20 \%$ interest. If $A ( t )$ is the amount of the investment at time $t$ for the case of continuous compounding, write a differential equation and an initial condition satisfied by $A ( t )$ .

Find the orthogonal trajectories of the family of curves. $y = k x ^ { 9 }$

Solve the differential equation. $4 \frac { d w } { d t } + 7 e ^ { t + w } = 0$

$y = C e ^ { 4 x ^ { 2 } }$ is the solution of the differential equation $y ^ { \prime } = 8 x y$ . Find the solution that satisfies the initial condition $y ( 1 ) = 1$ .

Solve the differential equation. Select the correct answer. $y ^ { t } = x ^ { 2 } e ^ { - \sin x } - y \cos x$

Determine whether the differential equation is linear. $y ^ { t } + 3 x ^ { 2 } y = 6 x ^ { 2 }$

Which of the following functions is a solution of the differential equation? $y ^ { tt } + 16 y ^ { t } + 64 y = 0$