Question 1

Find the orthogonal trajectories of the family of curves $x y = k$ . Then draw several members
of each family on the same coordinate plane.

Accepted Answer

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

Question 2

Solve the differential equation $\frac { d y } { d t } = t ( y - 3 )$ , $y ( 2 ) = 3$ . From your solution, and the value of $y ( 5 )$ .

Accepted Answer

A)  $- 2$ 
B) 2 
C) 5 
D) 0 
E)  $- 3$ 
F)3
G) $- 5$ 
H) $\frac { 1 } { 5 }$ 
A)  $- 2$ 
B) 2 
C) 5 
D) 0 
E)  $- 3$ 
F)3
G) $- 5$ 
H) $\frac { 1 } { 5 }$

Question 3

Which of the following equations is satisfied by the function $x = \sin 2 t$ ?

Accepted Answer

A)  $x ^ { \prime } - 2 t = 0$ 
B)  $x ^ { \prime } - 4 x t = 0$ 
C)  $x ^ { \prime \prime } + 4 x = 0$ 
D)  $x ^ { \prime \prime } + x = 0$ 
E)  $x ^ { \prime } - 4 x = 0$ 
F) $x ^ { \prime } + 4 x = 0$ 
G) $x ^ { \prime \prime } - 4 x = 0$ 
H) $x ^ { \prime \prime } - x = 0$ 
A)  $x ^ { \prime } - 2 t = 0$ 
B)  $x ^ { \prime } - 4 x t = 0$ 
C)  $x ^ { \prime \prime } + 4 x = 0$ 
D)  $x ^ { \prime \prime } + x = 0$ 
E)  $x ^ { \prime } - 4 x = 0$ 
F) $x ^ { \prime } + 4 x = 0$ 
G) $x ^ { \prime \prime } - 4 x = 0$ 
H) $x ^ { \prime \prime } - x = 0$

Question 4

Consider the differential equation $t ^ { 2 } \frac { d y } { d t } = \sqrt { y }$ .(a) Find the general solution to the differential equation.(b) Find the solution with the initial-value $y ( 1 ) = 0$ .(c) Find the solution with the initial-value $y ( 1 ) = 1$ .

Accepted Answer

The answer of Consider the differential equation \(t ^ {...

Question 5

A tank is filled with 100 L (liter) of water. Brine containing 0.4 kg of salt per liter is poured into the tank at 5 L per minute, and the well-stirred mixture leaves the tank at the same rate.(a) Write the differential equation that the amount of salt, $s ( t )$ in the tank must satisfy.(b) Find the amount of salt in the tank at any time, $t$ .(c) How much salt is present after 20 min?
(d) What is the concentration of solution in the tank after a long time?

Accepted Answer

(a) @#IMG-DLM& , @#IMG-DLM& (b

Question 6

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0002 P ^ { 2 }$ . Solve this differential equation with the initial condition P(0) = 20.

Accepted Answer

The answer of Suppose a population growth is modeled by...

Question 7

Find the solution of the initial-value problem $\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)$ , $y ( 0 ) = 0$ .

Accepted Answer

A)  $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right)$ 
C)  $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
E)  $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$ 
F) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right)$ 
G) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
H) $\frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
A)  $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$ 
B)  $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right)$ 
C)  $- \frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
D)  $\frac { 1 } { 2 } \cos \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
E)  $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) - \frac { 1 } { 2 }$ 
F) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right)$ 
G) $- \frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$ 
H) $\frac { 1 } { 2 } \sin \left( x ^ { 2 } \right) + \frac { 1 } { 2 }$

Question 8

The population of two species is modeled by the system of equations $$\left\{ \begin{array} { l } 
\frac { d x } { d t } = 2 x - x y \
\frac { d y } { d t } = - 4 y + x y
\end{array} \right.$$ .(a) Find an expression for $\frac { d y } { d x }$ .(b) A possible direction field for the differential equation in part (a) is given below:   Use this graph to sketch a phase portrait with each of P, Q, R, and S as an initial condition. Describe the behavior of the trajectories near the nonzero equilibrium solutions.(c) Graph x and y as function of t. What happens to the population of the two species as the time t increases without bound?

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& (c) @#IMG-DLM& The p

Question 9

Suppose $\frac { d y } { d x } = 2 x$ , $y ( 0 ) = 1$ . Use Euler's method with step size $h = 2$ to approximate $y ( 4 )$ .

Accepted Answer

A) 5 
B) 2 
C) 3 
D) 17 
E) 1
F)4
G)9
H)16 
A) 5 
B) 2 
C) 3 
D) 17 
E) 1
F)4
G)9
H)16

Question 10

Find the equation of a curve that passes through the point $( 0,7 )$ and whose slope at a point $( x , y )$ is $4 x ^ { 3 } y$ .

Accepted Answer

The answer of Find the equation of a curve that...

Question 11

Which of the following is a solution of the differential equation $\frac { d y } { d x } + 2 x y = x$ ?

Accepted Answer

A)  $y = 1 + e ^ { - x ^ { 2 } }$ 
B)  $y = e ^ { 2 x }$ 
C)  $y = x e ^ { - x ^ { 2 } }$ 
D)  $y = 1 + e ^ { - x ^ { 2 } }$ 
E)  $y = \frac { 1 } { 2 } + e ^ { - x ^ { 2 } }$ 
F) $y = e ^ { - x ^ { 2 } }$ 
G) $y = \frac { 1 } { 2 } + e ^ { x ^ { 2 } }$ 
H)None of these 
A)  $y = 1 + e ^ { - x ^ { 2 } }$ 
B)  $y = e ^ { 2 x }$ 
C)  $y = x e ^ { - x ^ { 2 } }$ 
D)  $y = 1 + e ^ { - x ^ { 2 } }$ 
E)  $y = \frac { 1 } { 2 } + e ^ { - x ^ { 2 } }$ 
F) $y = e ^ { - x ^ { 2 } }$ 
G) $y = \frac { 1 } { 2 } + e ^ { x ^ { 2 } }$ 
H)None of these

Question 12

A direction field is given below. Which of the following represents its differential equation?

Accepted Answer

A)  $\frac { d y } { d x } = \sin x$ 
B)  $\frac { d y } { d x } = - y$ 
C)  $\frac { d y } { d x } = y - \frac { 1 } { 2 } y ^ { 2 }$ 
D)  $\frac { d y } { d x } = x + y$ 
E)  $\frac { d y } { d x } = x ^ { 2 }$ 
F) $\frac { d y } { d x } = 1$ 
G) $\frac { d y } { d x } = y ^ { 2 }$ 
H) $\frac { d y } { d x } = x ^ { 2 } + y ^ { 2 }$ 
A)  $\frac { d y } { d x } = \sin x$ 
B)  $\frac { d y } { d x } = - y$ 
C)  $\frac { d y } { d x } = y - \frac { 1 } { 2 } y ^ { 2 }$ 
D)  $\frac { d y } { d x } = x + y$ 
E)  $\frac { d y } { d x } = x ^ { 2 }$ 
F) $\frac { d y } { d x } = 1$ 
G) $\frac { d y } { d x } = y ^ { 2 }$ 
H) $\frac { d y } { d x } = x ^ { 2 } + y ^ { 2 }$

Question 13

A direction field for a differential equation is given below. Use a straightedge to draw the graphs of the Euler approximations to the solution curve over the interval $[ 0,1.2 ]$ that passes through $y ( 0 ) = 1$ . Use as step sizes $h = 1.2$ , $h = 0.6$ , and $h = 0.2$ .

Accepted Answer

The answer of A direction field for a differential equation...

Question 14

In the presence of air resistance for an object in free fall, the velocity is the solution of the
differential equation $\frac { d v } { d t } = g - k v$ where $g$ is the constant of acceleration due to earth's gravity and $k$ is a positive constant.(a) Find the solution for the equation that satisfies the initial value that $v ( 0 ) = 0$ .(b) As time, $t$ , increases without bound, what is the limiting velocity. (Note: the limiting velocity during free fall is called the terminal velocity.)
(c) Sketch your solution from part (a).

Accepted Answer

The answer of In the presence of air resistance for...

Question 15

Suppose that a population grows according to a logistic model.(a) Write the differential equation for this situation with k = 0.01 and carrying capacity of 60 thousand.(b) Solve the differential equation in part (a) with the initial condition t = 0 (hours) and population P = 1 thousand.(c) Find the population for t = 10 hours, t = 100 hours, and t = 1000 hours.(d) After how many hours does the population reach 2 thousand? 30 thousand? 55 thousand?
(e) As the time t increases without bound, what happens to the population?
(f) Sketch the graph of the solution of the differential equation.

Accepted Answer

(a) @#IMG-DLM& (b) @#IMG-DLM& (c) @#IMG-DLM& (d) The population will r

Question 16

Suppose $\frac { d y } { d x } = 3 y - x$ , $y ( 0 ) = 1$ . Use Euler's method with step size $h = 0.5$ to approximate $y ( 1 )$ .

Accepted Answer

A) 10 
B) 7.5 
C) 9.5 
D) 5 
E) 1
F)2.5
G)4
H)6 
A) 10 
B) 7.5 
C) 9.5 
D) 5 
E) 1
F)2.5
G)4
H)6

Question 17

A population is modeled by the differential equation $\frac { d P } { d t } = 0.4 P \left( 1 - \frac { P } { 250 } \right)$ where $P ( t )$ is the population at time $t$ .(a) What are the equilibrium solutions?
(b) For what values of $P$ is the population increasing?
(c) For what values of $P$ is the population decreasing?
(d) Use the information from above to sketch the direction field for the given differential equation.

Accepted Answer

(a) @#IMG-DLM& , @#IMG-DLM& (b

Question 18

Consider the differential equation $\frac { d y } { d t } = t + 1$ .(a) What are the equilibrium solutions?
(b) Sketch the direction field for the given differential equation.

Accepted Answer

The answer of Consider the differential equation \(\frac { d...

Question 19

A direction field for a differential equation is given below:   
(a) Sketch the graphs of the solutions that have initial condition $P$ and initial condition $Q$ .
(b) Determine whether the differential equation is autonomous. Explain your answer.

Accepted Answer

(a) @#IMG-DLM& 
(b) It is auto

Question 20

Suppose a hail ball melts at a rate proportional to its surface area. If from $t = 0$ to $t = 5$ minutes its radius decreases from $r = 3$ cm to $r = 2$ cm, how long does it take to melt completely?

Accepted Answer

@#IMG-DLM& where @#IMG-DLM& represents the volume

Find the orthogonal trajectories of the family of curves $x y = k$ . Then draw several members of each family on the same coordinate plane.

Solve the differential equation $\frac { d y } { d t } = t ( y - 3 )$ , $y ( 2 ) = 3$ . From your solution, and the value of $y ( 5 )$ .

Which of the following equations is satisfied by the function $x = \sin 2 t$ ?

Consider the differential equation $t ^ { 2 } \frac { d y } { d t } = \sqrt { y }$ .(a) Find the general solution to the differential equation.(b) Find the solution with the initial-value $y ( 1 ) = 0$ .(c) Find the solution with the initial-value $y ( 1 ) = 1$ .

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0002 P ^ { 2 }$ . Solve this differential equation with the initial condition P(0) = 20.

Find the solution of the initial-value problem $\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)$ , $y ( 0 ) = 0$ .

Suppose $\frac { d y } { d x } = 2 x$ , $y ( 0 ) = 1$ . Use Euler's method with step size $h = 2$ to approximate $y ( 4 )$ .

Find the equation of a curve that passes through the point $( 0,7 )$ and whose slope at a point $( x , y )$ is $4 x ^ { 3 } y$ .

Which of the following is a solution of the differential equation $\frac { d y } { d x } + 2 x y = x$ ?

A direction field is given below. Which of the following represents its differential equation?

A direction field for a differential equation is given below. Use a straightedge to draw the graphs of the Euler approximations to the solution curve over the interval $[ 0,1.2 ]$ that passes through $y ( 0 ) = 1$ . Use as step sizes $h = 1.2$ , $h = 0.6$ , and $h = 0.2$ .

Suppose $\frac { d y } { d x } = 3 y - x$ , $y ( 0 ) = 1$ . Use Euler's method with step size $h = 0.5$ to approximate $y ( 1 )$ .

Consider the differential equation $\frac { d y } { d t } = t + 1$ .(a) What are the equilibrium solutions? (b) Sketch the direction field for the given differential equation.

A direction field for a differential equation is given below: (a) Sketch the graphs of the solutions that have initial condition $P$ and initial condition $Q$ . (b) Determine whether the differential equation is autonomous. Explain your answer.

Suppose a hail ball melts at a rate proportional to its surface area. If from $t = 0$ to $t = 5$ minutes its radius decreases from $r = 3$ cm to $r = 2$ cm, how long does it take to melt completely?

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Exam 7: Differential Equations

Find the orthogonal trajectories of the family of curves xy=kx y = kxy=k . Then draw several members of each family on the same coordinate plane.

Solve the differential equation dydt=t(y−3)\frac { d y } { d t } = t ( y - 3 )dtdy​=t(y−3) , y(2)=3y ( 2 ) = 3y(2)=3 . From your solution, and the value of y(5)y ( 5 )y(5) .

Which of the following equations is satisfied by the function x=sin⁡2tx = \sin 2 tx=sin2t ?

Suppose a population growth is modeled by the logistic equation dPdt=0.01P−0.0002P2\frac { d P } { d t } = 0.01 P - 0.0002 P ^ { 2 }dtdP​=0.01P−0.0002P2 . Solve this differential equation with the initial condition P(0) = 20.

Find the solution of the initial-value problem dydx=xsin⁡(x2)\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)dxdy​=xsin(x2) , y(0)=0y ( 0 ) = 0y(0)=0 .

Suppose dydx=2x\frac { d y } { d x } = 2 xdxdy​=2x , y(0)=1y ( 0 ) = 1y(0)=1 . Use Euler's method with step size h=2h = 2h=2 to approximate y(4)y ( 4 )y(4) .

Find the equation of a curve that passes through the point (0,7)( 0,7 )(0,7) and whose slope at a point (x,y)( x , y )(x,y) is 4x3y4 x ^ { 3 } y4x3y .

Which of the following is a solution of the differential equation dydx+2xy=x\frac { d y } { d x } + 2 x y = xdxdy​+2xy=x ?

A direction field is given below. Which of the following represents its differential equation?

Suppose dydx=3y−x\frac { d y } { d x } = 3 y - xdxdy​=3y−x , y(0)=1y ( 0 ) = 1y(0)=1 . Use Euler's method with step size h=0.5h = 0.5h=0.5 to approximate y(1)y ( 1 )y(1) .

Consider the differential equation dydt=t+1\frac { d y } { d t } = t + 1dtdy​=t+1 .(a) What are the equilibrium solutions? (b) Sketch the direction field for the given differential equation.

A direction field for a differential equation is given below: (a) Sketch the graphs of the solutions that have initial condition PPP and initial condition QQQ . (b) Determine whether the differential equation is autonomous. Explain your answer.

Suppose a hail ball melts at a rate proportional to its surface area. If from t=0t = 0t=0 to t=5t = 5t=5 minutes its radius decreases from r=3r = 3r=3 cm to r=2r = 2r=2 cm, how long does it take to melt completely?

Functions and Models

Limits and Derivatives

Differentiation Rules

Applications of Differentiation

Integrals

Applications of Integration

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Filters

Find the orthogonal trajectories of the family of curves $x y = k$ . Then draw several members of each family on the same coordinate plane.

Solve the differential equation $\frac { d y } { d t } = t ( y - 3 )$ , $y ( 2 ) = 3$ . From your solution, and the value of $y ( 5 )$ .

Which of the following equations is satisfied by the function $x = \sin 2 t$ ?

Suppose a population growth is modeled by the logistic equation $\frac { d P } { d t } = 0.01 P - 0.0002 P ^ { 2 }$ . Solve this differential equation with the initial condition P(0) = 20.

Find the solution of the initial-value problem $\frac { d y } { d x } = x \sin \left( x ^ { 2 } \right)$ , $y ( 0 ) = 0$ .

Suppose $\frac { d y } { d x } = 2 x$ , $y ( 0 ) = 1$ . Use Euler's method with step size $h = 2$ to approximate $y ( 4 )$ .

Find the equation of a curve that passes through the point $( 0,7 )$ and whose slope at a point $( x , y )$ is $4 x ^ { 3 } y$ .

Which of the following is a solution of the differential equation $\frac { d y } { d x } + 2 x y = x$ ?

Suppose $\frac { d y } { d x } = 3 y - x$ , $y ( 0 ) = 1$ . Use Euler's method with step size $h = 0.5$ to approximate $y ( 1 )$ .

Consider the differential equation $\frac { d y } { d t } = t + 1$ .(a) What are the equilibrium solutions? (b) Sketch the direction field for the given differential equation.

A direction field for a differential equation is given below: (a) Sketch the graphs of the solutions that have initial condition $P$ and initial condition $Q$ . (b) Determine whether the differential equation is autonomous. Explain your answer.

Suppose a hail ball melts at a rate proportional to its surface area. If from $t = 0$ to $t = 5$ minutes its radius decreases from $r = 3$ cm to $r = 2$ cm, how long does it take to melt completely?