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

The answer of Find the solution of the differential equation...

Question 2

Solve the initial-value problem.
$$r ^ { t } + 2 t r = r , \quad r ( 0 ) = 6$$

Accepted Answer

A) 
$$r ( t ) = \frac { e ^ { t ^ { 2 } } } { 6 }$$ 
B)  $r ( t ) = 6 e ^ { t - t ^ { 2 } }$ 
C)  $r ( t ) = e ^ { t - 6 t ^ { 2 } }$ 
D)  $r ( t ) = 6 e ^ { t - 2 t }$ 
E)  $r ( t ) = 6 e ^ { 2 t }$ 
A) 
$$r ( t ) = \frac { e ^ { t ^ { 2 } } } { 6 }$$ 
B)  $r ( t ) = 6 e ^ { t - t ^ { 2 } }$ 
C)  $r ( t ) = e ^ { t - 6 t ^ { 2 } }$ 
D)  $r ( t ) = 6 e ^ { t - 2 t }$ 
E)  $r ( t ) = 6 e ^ { 2 t }$

Question 3

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.
$$y ^ { t } = 5 x + y ^ { 2 } , y ( 0 ) = 0$$

Accepted Answer

The answer of Use Euler's method with step size $0.25$...

Question 4

Kirchhoff's Law gives us the derivative equation $Q ^ { \prime } = 12 - 4 Q$.
If $Q ( 0 ) = 0$, use Euler's method with step size $0.1$ to estimate $Q$ after $0.3$ second.

Accepted Answer

The answer of Kirchhoff's Law gives us the derivative equation...

Question 5

Let $P ( t )$ be the performance level of someone learning a skill as a function of the training time $t$. The graph of $P$ is called a learning curve. We propose the differential equation
$$\frac { d P } { d t } = r ( G - P ( t ) )$$
as a reasonable model for learning, where $r$ is a positive constant. Solve it as a linear differential equation.

Accepted Answer

The answer of Let $P ( t )$ be the...

Question 6

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. Select the correct answer.
$$y ^ { t } = 3 x + y ^ { 2 } , y ( 0 ) = 0$$

Accepted Answer

A)  $y ( 1 ) = 2.7154$ 
B)  $y ( 1 ) = 2.2154$ 
C)  $y ( 1 ) = 3.7154$ 
D)  $y ( 1 ) = 1.2154$ 
A)  $y ( 1 ) = 2.7154$ 
B)  $y ( 1 ) = 2.2154$ 
C)  $y ( 1 ) = 3.7154$ 
D)  $y ( 1 ) = 1.2154$

Question 7

Solve the differential equation.
$$x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$$

Accepted Answer

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

Question 8

Solve the differential equation. Select the correct answer.
$$7 y y ^ { t } = 5 x$$

Accepted Answer

A)  $7 x ^ { 2 } - 5 y ^ { 2 } = C$ 
B)  $5 x ^ { 2 } + 7 y ^ { 2 } = C$ 
C)  $5 x ^ { 2 } + 7 y ^ { 2 } = 12$ 
D)  $5 x ^ { 2 } - 7 y ^ { 2 } = C$ 
E)  $7 x ^ { 2 } + 5 y ^ { 2 } = C$ 
A)  $7 x ^ { 2 } - 5 y ^ { 2 } = C$ 
B)  $5 x ^ { 2 } + 7 y ^ { 2 } = C$ 
C)  $5 x ^ { 2 } + 7 y ^ { 2 } = 12$ 
D)  $5 x ^ { 2 } - 7 y ^ { 2 } = C$ 
E)  $7 x ^ { 2 } + 5 y ^ { 2 } = C$

Question 9

Determine whether the differential equation is linear.
$$y ^ { \prime } + y \ln x = x ^ { 2 } y ^ { 2 }$$

Accepted Answer

The answer of Determine whether the differential equation is linear.
\[y...

Question 10

A phase trajectory is shown for populations of rabbits $( R )$ and foxes $( F )$. Describe how each population changes as time goes by. Select the correct answer.

Select the correct statement.

Accepted Answer

A)  At $t = B$ the number of rabbits rebounds to 500 . 
B)  At $t = B$ the number of foxes reaches a maximum of about 2400 . 
C)  At $t = B$ the population of foxes reaches a minimum of about 30 . 
A)  At $t = B$ the number of rabbits rebounds to 500 . 
B)  At $t = B$ the number of foxes reaches a maximum of about 2400 . 
C)  At $t = B$ the population of foxes reaches a minimum of about 30 .

Question 11

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

Accepted Answer

A)  $y ( x ) = 3 x ( \ln x ) ^ { 2 }$ 
B) 
$y ( x ) = \frac { 1 } { 2 } x ( \ln x ) ^ { 2 } + 3 x$ 
C) 
$y ( x ) = 3 x \ln x + \frac { 1 } { 2 } x$ 
D) 
$y ( x ) = 3 \ln x + \frac { 1 } { 2 } x$ 
E) 
$y ( x ) = \frac { 1 } { 2 } ( \ln x ) ^ { 3 } + x$ 
A)  $y ( x ) = 3 x ( \ln x ) ^ { 2 }$ 
B) 
$y ( x ) = \frac { 1 } { 2 } x ( \ln x ) ^ { 2 } + 3 x$ 
C) 
$y ( x ) = 3 x \ln x + \frac { 1 } { 2 } x$ 
D) 
$y ( x ) = 3 \ln x + \frac { 1 } { 2 } x$ 
E) 
$y ( x ) = \frac { 1 } { 2 } ( \ln x ) ^ { 3 } + x$

Question 12

A tank contains $1050 \mathrm {~L}$ of brine with $20 \mathrm {~kg}$ of dissolved salt. Pure water enters the tank at a rate of $16 \mathrm {~L} / \mathrm { min }$. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 20 minutes?

Accepted Answer

The answer of A tank contains $1050 \mathrm {~L}$ of...

Question 13

A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the $\mathrm { y }$-coordinate $P$. What is the equation of the curve?

Accepted Answer

The answer of A curve passes through the point \((...

Question 14

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 50 cells. Find the number of cells after 6 hours.

Accepted Answer

A)  $P = 13,119,200$ 
B)  $P = 13,107,200$ 
C)  $P = 13,227,200$ 
D)  $P = 13,207,200$ 
E)  $P = 13,247,200$ 
A)  $P = 13,119,200$ 
B)  $P = 13,107,200$ 
C)  $P = 13,227,200$ 
D)  $P = 13,207,200$ 
E)  $P = 13,247,200$

Question 15

Find the orthogonal trajectories of the family of curves.
$$y = ( x + k ) ^ { - 8 }$$

Accepted Answer

The answer of Find the orthogonal trajectories of the family...

Question 16

Solve the differential equation.
$$7 y y ^ { t } = 5 x$$

Accepted Answer

The answer of Solve the differential equation.
\[7 y y ^...

Question 17

A sum of $\$ 3,000$ 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

The answer of A sum of $\$ 3,000$ is invested...

Question 18

Solve the initial-value problem.
$$r ^ { t } + 2 t r = r , r ( 0 ) = 4$$

Accepted Answer

The answer of Solve the initial-value problem.
\[r ^ { t...

Question 19

Solve the initial-value problem. Select the correct answer.
$$x y ^ { t } = y + x ^ { 2 } \sin x , \quad y ( 7 \pi ) = 0$$

Accepted Answer

A)  $y = x \sin x - 7 x$ 
B)  $y = 7 x \sin x$ 
C)  $y = - x \cos x - x$ 
D)  $y = 7 x \cos x - \sin x$ 
E)  $y = - x \sin x + 7 x$ 
A)  $y = x \sin x - 7 x$ 
B)  $y = 7 x \sin x$ 
C)  $y = - x \cos x - x$ 
D)  $y = 7 x \cos x - \sin x$ 
E)  $y = - x \sin x + 7 x$

Question 20

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

Accepted Answer

The answer of Solve the differential equation.
\[y ^ {t }...

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

Solve the initial-value problem. $r ^ { t } + 2 t r = r , \quad r ( 0 ) = 6$

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. $y ^ { t } = 5 x + y ^ { 2 } , y ( 0 ) = 0$

Kirchhoff's Law gives us the derivative equation $Q ^ { \prime } = 12 - 4 Q$ . If $Q ( 0 ) = 0$ , use Euler's method with step size $0.1$ to estimate $Q$ after $0.3$ second.

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. Select the correct answer. $y ^ { t } = 3 x + y ^ { 2 } , y ( 0 ) = 0$

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

Solve the differential equation. Select the correct answer. $7 y y ^ { t } = 5 x$

Determine whether the differential equation is linear. $y ^ { \prime } + y \ln x = x ^ { 2 } y ^ { 2 }$

A phase trajectory is shown for populations of rabbits $( R )$ and foxes $( F )$ . Describe how each population changes as time goes by. Select the correct answer. Select the correct statement.

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

A tank contains $1050 \mathrm {~L}$ of brine with $20 \mathrm {~kg}$ of dissolved salt. Pure water enters the tank at a rate of $16 \mathrm {~L} / \mathrm { min }$ . The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 20 minutes?

A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the $\mathrm { y }$ -coordinate $P$ . What is the equation of the curve?

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 50 cells. Find the number of cells after 6 hours.

Find the orthogonal trajectories of the family of curves. $y = ( x + k ) ^ { - 8 }$

Solve the differential equation. $7 y y ^ { t } = 5 x$

A sum of $\$ 3,000$ 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. $r ^ { t } + 2 t r = r , r ( 0 ) = 4$

Solve the initial-value problem. Select the correct answer. $x y ^ { t } = y + x ^ { 2 } \sin x , \quad y ( 7 \pi ) = 0$

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

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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

Solve the initial-value problem. rt+2tr=r,r(0)=6r ^ { t } + 2 t r = r , \quad r ( 0 ) = 6rt+2tr=r,r(0)=6

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=5x+y2,y(0)=0y ^ { t } = 5 x + y ^ { 2 } , y ( 0 ) = 0yt=5x+y2,y(0)=0

Kirchhoff's Law gives us the derivative equation Q′=12−4QQ ^ { \prime } = 12 - 4 QQ′=12−4Q . If Q(0)=0Q ( 0 ) = 0Q(0)=0 , use Euler's method with step size 0.10.10.1 to estimate QQQ after 0.30.30.3 second.

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. Select the correct answer. yt=3x+y2,y(0)=0y ^ { t } = 3 x + y ^ { 2 } , y ( 0 ) = 0yt=3x+y2,y(0)=0

Solve the differential equation. x2dydx−y=2x3e−1/xx ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }x2dxdy​−y=2x3e−1/x

Solve the differential equation. Select the correct answer. 7yyt=5x7 y y ^ { t } = 5 x7yyt=5x

Determine whether the differential equation is linear. y′+yln⁡x=x2y2y ^ { \prime } + y \ln x = x ^ { 2 } y ^ { 2 }y′+ylnx=x2y2

A phase trajectory is shown for populations of rabbits (R)( R )(R) and foxes (F)( F )(F) . Describe how each population changes as time goes by. Select the correct answer. Select the correct statement.

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

A curve passes through the point (8,2)( 8,2 )(8,2) and has the property that the slope of the curve at every point PPP is 3 times the y\mathrm { y }y -coordinate PPP . What is the equation of the curve?

A common inhabitant of human intestines is the bacterium Escherichia coli. A cell of this bacterium in a nutrient-broth medium divides into two cells every 20 minutes. The initial population of a culture is 50 cells. Find the number of cells after 6 hours.

Find the orthogonal trajectories of the family of curves. y=(x+k)−8y = ( x + k ) ^ { - 8 }y=(x+k)−8

Solve the differential equation. 7yyt=5x7 y y ^ { t } = 5 x7yyt=5x

A sum of $3,000\$ 3,000$3,000 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. rt+2tr=r,r(0)=4r ^ { t } + 2 t r = r , r ( 0 ) = 4rt+2tr=r,r(0)=4

Solve the initial-value problem. Select the correct answer. xyt=y+x2sin⁡x,y(7π)=0x y ^ { t } = y + x ^ { 2 } \sin x , \quad y ( 7 \pi ) = 0xyt=y+x2sinx,y(7π)=0

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

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Techniques of Integration

Further Applications of Integration

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

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

Solve the initial-value problem. $r ^ { t } + 2 t r = r , \quad r ( 0 ) = 6$

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. $y ^ { t } = 5 x + y ^ { 2 } , y ( 0 ) = 0$

Kirchhoff's Law gives us the derivative equation $Q ^ { \prime } = 12 - 4 Q$ . If $Q ( 0 ) = 0$ , use Euler's method with step size $0.1$ to estimate $Q$ after $0.3$ second.

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. Select the correct answer. $y ^ { t } = 3 x + y ^ { 2 } , y ( 0 ) = 0$

Solve the differential equation. $x ^ { 2 } \frac { d y } { d x } - y = 2 x ^ { 3 } e ^ { - 1 / x }$

Solve the differential equation. Select the correct answer. $7 y y ^ { t } = 5 x$

Determine whether the differential equation is linear. $y ^ { \prime } + y \ln x = x ^ { 2 } y ^ { 2 }$

A phase trajectory is shown for populations of rabbits $( R )$ and foxes $( F )$ . Describe how each population changes as time goes by. Select the correct answer. Select the correct statement.

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

A curve passes through the point $( 8,2 )$ and has the property that the slope of the curve at every point $P$ is 3 times the $\mathrm { y }$ -coordinate $P$ . What is the equation of the curve?

Find the orthogonal trajectories of the family of curves. $y = ( x + k ) ^ { - 8 }$

Solve the differential equation. $7 y y ^ { t } = 5 x$

A sum of $\$ 3,000$ 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. $r ^ { t } + 2 t r = r , r ( 0 ) = 4$

Solve the initial-value problem. Select the correct answer. $x y ^ { t } = y + x ^ { 2 } \sin x , \quad y ( 7 \pi ) = 0$

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