Question 1

Solve the problem.
-The system of equations  dxdt=(−2+3y) x\frac { d x } { d t } = ( - 2 + 3 y )  xdtdx​=(−2+3y) x  and  dydt=(−2+x) y\frac { d y } { d t } = ( - 2 + x )  ydtdy​=(−2+x) y  describes the growth rates of two symbiotic (dependent)  species of animals (such as the rhinoceros and a type of bird which eats insects from its back) . What is necessarily true of the two populations at the equilibrium points?

Accepted Answer

A)   They are both at a maximum.
B)   Both populations equal zero.
C)   They are both at a minimum.
D)   They both remain constant over all time.

Question 2

Obtain a slope field and add to its graphs of the solution curves passing through the given points.
-$$y ^ { \prime } = 3 ( y - 1 ) \text { with } ( 2,0 )$$

Accepted Answer

The answer of Obtain a slope field and add to...

Question 3

Solve the problem.
-A 200 gal tank is half full of distilled water. At time = 0, a solution containing 1 lb/gal of concentrate enters the tank at the rate of 4 gal/min, and the well-stirred mixture is withdrawn at the rate of 2 gal/min. When the tank
Is full, how many pounds of concentrate will it contain?

Accepted Answer

A)   100 pounds
B)   200 pounds
C)   120 pounds
D)   150 pounds

Question 4

The autonomous differential equation represents a model for population growth. Use phase line analysis to sketch solution curves for P(t), selecting different starting values P(0). Which equilibria are stable, and which are unstable?
-$\frac { \mathrm { dP } } { \mathrm { dt } } = \mathrm { P } ( \mathrm { P } - 5 )$

Accepted Answer

has a stable equil...

Question 5

Determine which of the following equations is correct.
- 1sin⁡x∫sin⁡xdx=\frac { 1 } { \sin x } \int \sin x d x =sinx1​∫sinxdx=

Accepted Answer

A)    1/sin⁡x+C1 / \sin x + C1/sinx+C 
B)    −cot⁡x+C- \cot x + C−cotx+C 
C)    cos⁡x+C\cos x + Ccosx+C 
D)    sin⁡x+C\sin x + Csinx+C

Question 6

Solve the differential equation.
- 3x2y′−2xy=y−33 x ^ { 2 } y ^ { \prime } - 2 x y = y ^ { - 3 }3x2y′−2xy=y−3

Accepted Answer

A)    y4=−411x+Cx8/3y ^ { 4 } = \frac { - 4 } { 11 x } + C x ^ { 8 / 3 }y4=11x−4​+Cx8/3 
B)    y4=−411x+Cx11/3y ^ { 4 } = \frac { - 4 } { 11 x } + C x ^ { 11 / 3 }y4=11x−4​+Cx11/3 
C)    y2=−411x+Cx11/3y ^ { 2 } = \frac { - 4 } { 11 x } + C x ^ { 11 / 3 }y2=11x−4​+Cx11/3 
D)    y2=−411x+Cx8/3y ^ { 2 } = \frac { - 4 } { 11 x } + C x ^ { 8 / 3 }y2=11x−4​+Cx8/3

Question 7

Solve the differential equation.
-  0">exdydx+3exy=2,x>0e ^ { x } \frac { d y } { d x } + 3 e ^ { x } y = 2 , x > 0exdxdy​+3exy=2,x>0

Accepted Answer

A)     0">y=e−3x+Ce−x,x>0y = e ^ { - 3 x } + C e ^ { - x } , x > 0y=e−3x+Ce−x,x>0 
B)     0">y=e−x+e−3x,x>0y = e ^ { - x } + e ^ { - 3 x } , x > 0y=e−x+e−3x,x>0 
C)     0">y=e−x+Ce−3x,x>0y = e ^ { - x } + C e ^ { - 3 x } , x > 0y=e−x+Ce−3x,x>0 
D)     0">y=ex+Ce−3x,x>0y = e ^ { x } + C e ^ { - 3 x } , x > 0y=ex+Ce−3x,x>0

Question 8

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places.
- y′=4xex4,y(1) =3,dx=0.1y ^ { \prime } = 4 x e ^ { x ^ { 4 } } , y ( 1 )  = 3 , d x = 0.1y′=4xex4,y(1) =3,dx=0.1

Accepted Answer

A)   y1 = 3.2699, y2 = 5.1723, y3 = 8.9899
B)   y1 = 4.0873, y2 = 5.9897, y3 = 9.8074
C)   y1 = 3.6786, y2 = 5.5810, y3 = 9.3986
D)   y1 = 4.4960, y2 = 6.3985, y3 = 10.2161  B

Question 9

Solve the initial value problem.
- dydx+xy=4x;y(0) =−4\frac { d y } { d x } + x y = 4 x ; y ( 0 )  = - 4dxdy​+xy=4x;y(0) =−4

Accepted Answer

A)    y=4ex2/2−8y = 4 e ^ { x ^ { 2 } / 2 } - 8y=4ex2/2−8 
B)    y=4e−x2/2−8y = 4 e ^ { - x ^ { 2 } / 2 } - 8y=4e−x2/2−8 
C)    y=−8ex2/2+4y = - 8 e ^ { x ^ { 2 } / 2 } + 4y=−8ex2/2+4 
D)    y=−8e−x2/2+4y = - 8 e ^ { - x ^ { 2 } / 2 } + 4y=−8e−x2/2+4 D

Question 10

Sketch several solution curves.
-$\frac { d y } { d x } = y ^ { 2 } - 1$

Accepted Answer

The answer of Sketch several solution curves.
-\(\frac { d y...

Question 11

Identify equilibrium values and determine which are stable and which are unstable.
- dydx=y2−16\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { y } ^ { 2 } - 16dxdy​=y2−16

Accepted Answer

A)   y = 4 is a stable equilibrium value and y = -4 is an unstable equilibrium.
B)   y = -4 and y = 5 are stable equilibrium values.
C)   y = -4 is a stable equilibrium value and y = 4 is an unstable equilibrium.
D)   There are no equilibrium values.

Question 12

Identify equilibrium values and determine which are stable and which are unstable.
-y = (y - 4) (y - 6) (y - 7)

Accepted Answer

A)   y = 7 is a stable equilibrium value and y = 6 and y = 4 are unstable equilibria.
B)   y = 4 is a stable equilibrium value and y = 6 and y = 7 are unstable equilibria.
C)   y = 6 is a stable equilibrium value and y = 4 and y = 7 are unstable equilibria.
D)   y = 5, y = 4 and y = 7 are unstable equilibria.

Question 13

Solve the problem.
-dy/dt = ky + f(t)  is a population model where y is the population at time t and f(t)  is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y
When f(t)  = 11t.

Accepted Answer

A)    y=550t+27,500+37,500e−.02t\mathrm { y } = 550 \mathrm { t } + 27,500 + 37,500 \mathrm { e } ^ { - .02 \mathrm { t } }y=550t+27,500+37,500e−.02t 
B)    y=550t−27,500+37,500e−.02ty = 550 \mathrm { t } - 27,500 + 37,500 \mathrm { e } ^ { - .02 \mathrm { t } }y=550t−27,500+37,500e−.02t 
C)    y=−550t−27,500+37,500e−.02ty = - 550 t - 27,500 + 37,500 e ^ { - .02 t }y=−550t−27,500+37,500e−.02t 
D)    y=−550t−27,500+37,500e⋅02ty = - 550 \mathrm { t } - 27,500 + 37,500 \mathrm { e }^{ \cdot 02 \mathrm { t }}y=−550t−27,500+37,500e⋅02t

Question 14

Solve the problem.
-dy/dt = ky + f(t)  is a population model where y is the population at time t and f(t)  is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y
When f(t)  = -8t.

Accepted Answer

A)    y=−400t−20,000−10,000e⋅02ty = - 400 t - 20,000 - 10,000 \mathrm { e } \cdot 02 \mathrm { t }y=−400t−20,000−10,000e⋅02t 
B)    y=−400t+20,000−10,000e−.02ty = - 400 t + 20,000 - 10,000 \mathrm { e } ^ { - .02 t }y=−400t+20,000−10,000e−.02t 
C)    y=400t+20,000−10,000e−.02ty = 400 t + 20,000 - 10,000 e ^ { - .02 t }y=400t+20,000−10,000e−.02t 
D)    y=400t+20,000−10,000e⋅02ty = 400 t + 20,000 - 10,000 \mathrm { e } ^{\cdot 02 \mathrm { t }}y=400t+20,000−10,000e⋅02t

Question 15

Find the orthogonal trajectories of the family of curves. Sketch several members of each family.
-y = -mx

Accepted Answer

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

Question 16

Construct a phase line. Identify signs of y and y .
- dydx=(y+1) (y−2) \frac{d y}{d x}=(y+1) (y-2) dxdy​=(y+1) (y−2)

Accepted Answer

A)  
B)  
C)  
D)

Question 17

Obtain a slope field and add to its graphs of the solution curves passing through the given points.
-$$\mathrm { y } ^ { \prime } = \frac { 2 \mathrm { y } } { \mathrm { x } } \text { with } ( - 2,0 )$$

Accepted Answer

The answer of Obtain a slope field and add to...

Question 18

Solve.
-Solve the initial value problem  dPdt=7kP2,P(0) =P0\frac { \mathrm { dP } } { \mathrm { dt } } = 7 \mathrm { kP } ^ { 2 } , \mathrm { P } ( 0 )  = \mathrm { P } _ { 0 }dtdP​=7kP2,P(0) =P0​

Accepted Answer

A)    P(t) =7P01−7kP0tP ( t )  = \frac { 7 P _ { 0 } } { 1 - 7 \mathrm { kP } _ { 0 } t }P(t) =1−7kP0​t7P0​​ 
B)    P(t) =P01−7kP0t\mathrm { P } ( \mathrm { t } )  = \frac { \mathrm { P } _ { 0 } } { 1 - 7 \mathrm { kP } _ { 0 } \mathrm { t } }P(t) =1−7kP0​tP0​​ 
C)    P(t) =P07−7kP0t\mathrm { P } ( \mathrm { t } )  = \frac { \mathrm { P } _ { 0 } } { 7 - 7 \mathrm { kP } _ { 0 } \mathrm { t } }P(t) =7−7kP0​tP0​​ 
D)    P(t) =P01−kP0t\mathrm { P } ( \mathrm { t } )  = \frac { \mathrm { P } _ { 0 } } { 1 - \mathrm { kP } _ { 0 } \mathrm { t } }P(t) =1−kP0​tP0​​

Question 19

Solve the problem.
-An office contains  1000ft31000 \mathrm { ft } ^ { 3 }1000ft3  of air initially free of carbon monoxide. Starting at time  =0= 0=0 , cigarette smoke containing  4%4 \%4%  carbon monoxide is blown into the room at the rate of  0.5ft3/min0.5 \mathrm { ft } 3 / \mathrm { min }0.5ft3/min . A ceiling fan keeps the air in the room well circulated and the air leaves the room at the same rate of  0.5ft3/min. Find 0.5 \mathrm { ft } ^ { 3 } / \mathrm { min } ^ { 	ext {. Find } }0.5ft3/min. Find   the time when the concentration of carbon monoxide reaches  0.01%0.01 \%0.01% .

Accepted Answer

A)   5.01 min
B)   6.01 min
C)   8.01 min
D)   7.01 min  A

Question 20

Solve the problem.
-How many seconds after the switch in an RL circuit is closed will it take the current i to reach 20% of its steady state value? Express answer in terms of R and L and round coefficient to the nearest hundredth.

Accepted Answer

A)   0.22 L/R seconds
B)   1.81 L/R seconds
C)   1.61 L/R seconds
D)   0.42 L/R seconds

Exam 10: First-Order Differential Equations

Correct Answer
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Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $y ^ { \prime } = 3 ( y - 1 ) \text { with } ( 2,0 )$

Correct Answer
verified

Correct Answer
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Correct Answer
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has a stable equil...
View Answer

Determine which of the following equations is correct. - $\frac { 1 } { \sin x } \int \sin x d x =$

Correct Answer
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Solve the differential equation. - $3 x ^ { 2 } y ^ { \prime } - 2 x y = y ^ { - 3 }$

Correct Answer
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Solve the differential equation. - $e ^ { x } \frac { d y } { d x } + 3 e ^ { x } y = 2 , x > 0$

Correct Answer
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Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. - $y ^ { \prime } = 4 x e ^ { x ^ { 4 } } , y ( 1 ) = 3 , d x = 0.1$

Correct Answer
verified
B

Solve the initial value problem. - $\frac { d y } { d x } + x y = 4 x ; y ( 0 ) = - 4$

Correct Answer
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D

Sketch several solution curves. - $\frac { d y } { d x } = y ^ { 2 } - 1$

Correct Answer
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Identify equilibrium values and determine which are stable and which are unstable. - $\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { y } ^ { 2 } - 16$

Correct Answer
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Identify equilibrium values and determine which are stable and which are unstable. -y = (y - 4) (y - 6) (y - 7)

Correct Answer
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Solve the problem. -dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y When f(t) = 11t.

Correct Answer
verified

Solve the problem. -dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y When f(t) = -8t.

Correct Answer
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Find the orthogonal trajectories of the family of curves. Sketch several members of each family. -y = -mx

Correct Answer
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Construct a phase line. Identify signs of y and y . - $\frac{d y}{d x}=(y+1) (y-2)$

Correct Answer
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Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $\mathrm { y } ^ { \prime } = \frac { 2 \mathrm { y } } { \mathrm { x } } \text { with } ( - 2,0 )$

Correct Answer
verified

Solve. -Solve the initial value problem $\frac { \mathrm { dP } } { \mathrm { dt } } = 7 \mathrm { kP } ^ { 2 } , \mathrm { P } ( 0 ) = \mathrm { P } _ { 0 }$

Correct Answer
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Correct Answer
verified
A

Solve the problem. -How many seconds after the switch in an RL circuit is closed will it take the current i to reach 20% of its steady state value? Express answer in terms of R and L and round coefficient to the nearest hundredth.

Correct Answer
verified

Exam 10: First-Order Differential Equations

Correct AnswerverifiedShow Answer

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - y′=3(y−1) with (2,0)y ^ { \prime } = 3 ( y - 1 ) \text { with } ( 2,0 )y′=3(y−1) with (2,0)

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Correct Answerverified has a stable equil...View AnswerShow Answer

Determine which of the following equations is correct. - 1sin⁡x∫sin⁡xdx=\frac { 1 } { \sin x } \int \sin x d x =sinx1​∫sinxdx=

Correct AnswerverifiedShow Answer

Solve the differential equation. - 3x2y′−2xy=y−33 x ^ { 2 } y ^ { \prime } - 2 x y = y ^ { - 3 }3x2y′−2xy=y−3

Correct AnswerverifiedShow Answer

Solve the differential equation. - exdydx+3exy=2,x>0e ^ { x } \frac { d y } { d x } + 3 e ^ { x } y = 2 , x > 0exdxdy​+3exy=2,x>0

Correct AnswerverifiedShow Answer

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. - y′=4xex4,y(1) =3,dx=0.1y ^ { \prime } = 4 x e ^ { x ^ { 4 } } , y ( 1 ) = 3 , d x = 0.1y′=4xex4,y(1) =3,dx=0.1

Correct AnswerverifiedB

Solve the initial value problem. - dydx+xy=4x;y(0) =−4\frac { d y } { d x } + x y = 4 x ; y ( 0 ) = - 4dxdy​+xy=4x;y(0) =−4

Correct AnswerverifiedD

Sketch several solution curves. - dydx=y2−1\frac { d y } { d x } = y ^ { 2 } - 1dxdy​=y2−1

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Identify equilibrium values and determine which are stable and which are unstable. - dydx=y2−16\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { y } ^ { 2 } - 16dxdy​=y2−16

Correct AnswerverifiedShow Answer

Identify equilibrium values and determine which are stable and which are unstable. -y = (y - 4) (y - 6) (y - 7)

Correct AnswerverifiedShow Answer

Solve the problem. -dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y When f(t) = 11t.

Correct AnswerverifiedShow Answer

Solve the problem. -dy/dt = ky + f(t) is a population model where y is the population at time t and f(t) is some function to describe the net effect on the population. Assume k = .02 and y = 10,000 when t = 0. Solve the differential equation of y When f(t) = -8t.

Correct AnswerverifiedShow Answer

Find the orthogonal trajectories of the family of curves. Sketch several members of each family. -y = -mx

Correct AnswerverifiedShow Answer

Construct a phase line. Identify signs of y and y . - dydx=(y+1) (y−2) \frac{d y}{d x}=(y+1) (y-2) dxdy​=(y+1) (y−2)

Correct AnswerverifiedShow Answer

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - y′=2yx with (−2,0)\mathrm { y } ^ { \prime } = \frac { 2 \mathrm { y } } { \mathrm { x } } \text { with } ( - 2,0 )y′=x2y​ with (−2,0)

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Solve. -Solve the initial value problem dPdt=7kP2,P(0) =P0\frac { \mathrm { dP } } { \mathrm { dt } } = 7 \mathrm { kP } ^ { 2 } , \mathrm { P } ( 0 ) = \mathrm { P } _ { 0 }dtdP​=7kP2,P(0) =P0​

Correct AnswerverifiedShow Answer

Correct AnswerverifiedA

Solve the problem. -How many seconds after the switch in an RL circuit is closed will it take the current i to reach 20% of its steady state value? Express answer in terms of R and L and round coefficient to the nearest hundredth.

Correct AnswerverifiedShow Answer

Correct Answer
verified

Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $y ^ { \prime } = 3 ( y - 1 ) \text { with } ( 2,0 )$

Correct Answer
verified

Correct Answer
verified

Correct Answer
verified

has a stable equil...
View Answer

Determine which of the following equations is correct. - $\frac { 1 } { \sin x } \int \sin x d x =$

Correct Answer
verified

Solve the differential equation. - $3 x ^ { 2 } y ^ { \prime } - 2 x y = y ^ { - 3 }$

Correct Answer
verified

Solve the differential equation. - $e ^ { x } \frac { d y } { d x } + 3 e ^ { x } y = 2 , x > 0$

Correct Answer
verified

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places. - $y ^ { \prime } = 4 x e ^ { x ^ { 4 } } , y ( 1 ) = 3 , d x = 0.1$

Correct Answer
verified
B

Solve the initial value problem. - $\frac { d y } { d x } + x y = 4 x ; y ( 0 ) = - 4$

Correct Answer
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D

Sketch several solution curves. - $\frac { d y } { d x } = y ^ { 2 } - 1$

Correct Answer
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Identify equilibrium values and determine which are stable and which are unstable. - $\frac { \mathrm { dy } } { \mathrm { dx } } = \mathrm { y } ^ { 2 } - 16$

Correct Answer
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Correct Answer
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Correct Answer
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Correct Answer
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Correct Answer
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Construct a phase line. Identify signs of y and y . - $\frac{d y}{d x}=(y+1) (y-2)$

Correct Answer
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Obtain a slope field and add to its graphs of the solution curves passing through the given points. - $\mathrm { y } ^ { \prime } = \frac { 2 \mathrm { y } } { \mathrm { x } } \text { with } ( - 2,0 )$

Correct Answer
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Solve. -Solve the initial value problem $\frac { \mathrm { dP } } { \mathrm { dt } } = 7 \mathrm { kP } ^ { 2 } , \mathrm { P } ( 0 ) = \mathrm { P } _ { 0 }$

Correct Answer
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Correct Answer
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A

Correct Answer
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