Tom Robbins WW Prob Lib1 WeBWorK, Version 1.7 - Demo Course · Tom Robbins WW Prob Lib1 WeBWorK,...

Tom Robbins WW Prob Lib1 WeBWorK, Version 1.7 - Demo CourseWeBWorK problems.

1.(1 pt) Match each of the following differential equationswith a solution from the list below.

1. y� ��

y � 02. y

� ��8y��

15y � 03. y

� ��8y��

15y � 04. 2x2y

� � �3xy

� � y

A. y � cos � x �B. y � e3x

C. y � e 5x

D. y � 1x

2.(1 pt) Match each of the differential equation with its solu-tion.

1. y� ��

13y��

42y � 02. 2x2y

� ��3xy

� � y3. xy

��y � x2

4. y� � �

y � 0

A. y � 3x�

x2

B. y � sin � x �C. y � x

12

D. y � e 7x

3.(1 pt) Match each differential equation to a function whichis a solution.FUNCTIONSA. y � 3x

�x2,

B. y � e 8x,C. y � sin � x � ,D. y � x

12 ,

E. y � 7exp � 5x � ,DIFFERENTIAL EQUATIONS

1. y� ��

10y��

16y � 02. y

� � 5y3. 2x2y

� ��3xy

� � y4. y

� ��y � 0

4.(1 pt) Match the following differential equations with theirsolutions.The symbols A, B, C in the solutions stand for arbitrary con-stants.You must get all of the answers correct to receive credit.

1. d2ydx2

�4y � 0

2. dydx � 2xy

x2 2y2

3. d2ydx2

�12 dy

dx

�36y � 0

4. dydx � 4xy

5. dydx

�18x2y � 18x2

A. 3yx2 � 2y3 � CB. y � Ae2x2

C. y � Ce 6x3 �1

D. y � Acos � 2x � � Bsin � 2x �E. y � Ae 6x � Bxe 6x

5.(1 pt)Just as there are simultaneous algebraic equations (where a

pair of numbers have to satisfy a pair of equations) there are sys-tems of differential equations, (where a pair of functions have tosatisfy a pair of differential equations).

Indicate which pairs of functions satisfy this system. It willtake some time to make all of the calculations.

y�1 � 5

2y1� 3

2y2 y

�2 � � 3

2y1� 5

2y2� A. y1 � e x y2 � e x� B. y1 � sin � x � y2 � cos � x �� C. y1 � sin � x � � cos � x � y2 � cos � x � � sin � x �� D. y1 � e4x y2 � �

e4x� E. y1 � 2e 2x y2 � 3e 2x� F. y1 � cos � x � y2 � �sin � x �� G. y1 � ex y2 � ex

As you can see, finding all of the solutions, particularly of asystem of equations, can be complicated and time consuming.It helps greatly if we study the structure of the family of solu-tions to the equations. Then if we find a few solutions we willbe able to predict the rest of the solutions using the structure ofthe family of solutions.

6.(1 pt) It can be helpful to classify a differential equation,so that we can predict the techniques that might help us to finda function which solves the equation. Two classifications arethe order of the equation – (what is the highest number ofderivatives involved) and whether or not the equation is linear .

Linearity is important because the structure of the the familyof solutions to a linear equation is fairly simple. Linear equa-tions can usually be solved completely and explicitly.

Determine whether or not each equation is linear:

? 1. � 1 � y2 � d2ydt2

�t dy

dt

�y � et

? 2. t2 d2ydt2

�t dy

dt

�2y � sin t

? 3. dydt

�ty2 � 0

? 4. d2ydt2

�sin � t � y � � sin t

7.(1 pt) It is easy to check that for any value of c, the functiony � x2

� cx2 is solution of equation xy

��2y � 4x2 � � x � 0 � . Find

the value of c for which the solution satisfies the initial conditiony � 3 � � 2.

c �8.(1 pt) The functions

y � x2 � cx2

are all solutions of equation:

xy� �

2y � 4x2 � � x � 0 �1

Find the constant c which produces a solution which also satis-fies the initial condition y � 2 � � 5.

c �9.(1 pt) It is easy to check that for any value of c, the function

y � ce 2x � e x is solution of equation y� �

2y � e x. Find thevalue of c for which the solution satisfies the initial conditiony � 2 � � 9.

c �10.(1 pt) The family of functions y � ce 2x

�e x is solution

of the equationy� �

2y � e x

Find the constant c which defines the solution which also satis-fies the initial condition y � 2 � � 2.

c �11.(1 pt) Find the two values of k for which y � x �� ekx is a

solution of the differential equationy� ��

12y��

27y � 0.smaller value =

larger value =

12.(1 pt) Some curves in the first quadrant have equationsy � Aexp � 2x � � where A is a positive constant.Different values of A give different curves. The curves form afamily, F.Let P �� 5 � 4 �� Let C be the member of the family F

that goes through P.A. Let y � f � x � be the equation of C. Find f � x � .f � x ��

B. Find the slope at P of the tangent to C.slope:C. A curve D is perpendicular to C at P . What is the slope ofthe tangent to D at the point P ? slope:D. Give a formula g � y � for the slope at � x � y � of the member of Fthat goes through � x � y � . The formula should not involve A or x.g � y � �E. A curve which at each of its points is perpendicular to themember of the family F that goes through the point is called anorthogonal trajectory to F. Each orthogonal trajectory to F satis-fies the differential equationdydx � � 1

g � y � � where g � y � is the answer to part D.

Find a function h � y � such that x � h � y � is the equation of theorthogonal trajectory to F that passes through the point P.h � y � �

13.(1 pt) The solution of a certain differential equation is ofthe formy � t � � aexp � 5t � � bexp � 9t � � where a and b are constants.The solution has initial conditions y � 0 � � 4 and y

� � 0 � � 2 �Find the solution by using the initial conditions to get linearequations for a and b �

y � t � �

Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c�

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1.(1 pt)

Match the following equations with their direction field. Clicking on each picture will give you an enlarged view. While you canprobably solve this problem by guessing, it is useful to try to predict characteristics of the direction field and then match them to thepicture.

Here are some handy characteristics to start with – you will develop more as you practice.

A. Set y equal to zero and look at how the derivative behaves along the x axis.B. Do the same for the y axis by setting x equal to 0C. Consider the curve in the plane defined by setting y’=0 – this should correspond to the points in the picture where the slope

is zero.D. Setting y’ equal to a constant other than zero gives the curve of points where the slope is that constant. These are called

isoclines, and can be used to construct the direction field picture by hand.

Go to this page to launch the phase plane plotter to check your answers. (Choose the ”integral curves utility” from the appletmenu, enter dx/dt=1 to identify the variables x and t and then enter the function you want for dy/dx = dy/dt = .....).

(You can also login as practice1, or practice2 (use the login name as a password) and you can then practice more versions of thisproblem and the next one.)

1. y� � 3sin � x � � 1

�y

2. y� � � 2x � y �� 2y �

3. y� � �

1�

2y

A B C

2.(1 pt)

This problem is harder, and doesn’t give you clues as to which matches you have right. Study the previous problem, if you arehaving trouble.

Go to this page to launch the phase plane plotter to check your answers.(You can also login as practice1, or practice2 (use the login name as a password) and you can then practice more versions of thisproblem and the previous one.)

Match the following equations with their direction field. Clicking on each picture will give you an enlarged view.

1. y� � xe 2x

�2y

2. y� � y � 4 � y �

3. y� � y3

6

�y� x3

64. y

� � 2x�

1�

y2

1

A B

C D

3.(1 pt)




is zero.2

D. Setting y’ equal to a constant other than zero gives the curve of points where the slope is that constant. These are calledisoclines, and can be used to construct the direction field picture by hand.

1. y� � 2xy

�2xe x2

2. y� � 2sin � 3x � � 1

�y

3. y� � y

x

�3cos � 2x �

4. y� � 2y

�2

A B

C D

4.(1 pt)




is zero.D. Setting y’ equal to a constant other than zero gives the curve of points where the slope is that constant. These are called

isoclines, and can be used to construct the direction field picture by hand.

1. y� � 2y

�x2e2x

2. y� � y

�2

3. y� � �

2�

x�

y3

4. y� � � 2x � y �� 2y �

A B

C D


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1.(1 pt) Find the particular solution of the differential equa-tion

dydx

�� x � 4 � e 2y

satisfying the initial condition y � 4 � � ln � 4 � .Answer: y=Your answer should be a function of x.


x2

y2�

6dydx

� 12y

satisfying the initial condition y � 1 � �� 7.Answer: y=Your answer should be a function of x.

3.(1 pt) Find u from the differential equation and initial con-dition.

dudt � e2 � 8t 2 � 7u � u � 0 � � 3 � 7 �u �4.(1 pt) Solve the separable differential equation for u.dudt � e5u � 8t

Use the following initial condition: u � 0 � � 10u �5.(1 pt) Solve the separable differential equation for u.dudt � e5u � 8t

Use the following initial condition: u � 0 � � �13

u �6.(1 pt) Solve the separable differential equation8x�

3y � x2�

1dydx � 0

Subject to the initial condition: y � 0 � � 0y �

(function of x only)

7.(1 pt) Find f � x � if y � f � x � satisfiesdydx � 130yx12

and the y intercept of the curve y � f � x � is 2.f � x ��8.(1 pt) Find an equation of the curve that satisfies.dydx � 72yx7

and whose y intercept is 6.y � x � �

(function of x)

9.(1 pt) Find the solution of the differential equation

3e4x dydx

� �16

xy2

which satisfies the initial condition y � 0 �� 1y =

10.(1 pt) Find a function y of x such that4yy

� � x and y � 4 � � 10 �y � (function of x)

11.(1 pt) Solve the seperable differential equation.

3yy� � x

Use the following initial condition: y � 3 � � 5x2 � (function of y)

12.(1 pt) Solve the differential equation� y11x � dydx � 1

�x �

Use the initial condition y � 1 � � 2 �Express y12 in terms of x �y12 �( function of x)

13.(1 pt) Solve the seperable differential equation for.dydx � 1 � x

xy6 ;x � 0

Use the following initial condition: y � 1 � � 4y7 �

( function of x)

14.(1 pt) Find the function y � y � x � (for x � 0 ) which satis-fies the separable differential equation

dydx � 8 � 14x

xy2 ;x � 0

with the initial condition: y � 1 � � 4y �

( function of x only)

15.(1 pt) Find the solution of the differential equation� ln � y �� 3 dydx

� x3y

which satisfies the initial condition y � 1 �� e2

y =

16.(1 pt) A. Solve the following initial value problem:� t2 � 24t�

95 � dydt

� y

with y � 12 � � 1. (Find y as a function of t.)y �

B. On what interval is the solution valid?Answer: It is valid for � t � .

C. Find the limit of the solution as t approaches the left endof the interval.(Your answer should be a number or the word ”infinite”.)Answer C:

D. Similar to C, but for the right end.Answer D:

17.(1 pt) The differential equationdydx

� cos � x � � y2 � 12y�

32 �� 6y�

40 �has an implicit general solution of the form F � x � y �� K �In fact, because the differential equation is separable, we can

define the solution curve implicitly by a function in the formF � x � y �� G � x � � H � y � � K

Find such a solution and then give the related functions re-quested.

F � x � y �� G � x � � H � y � �1

18.(1 pt) The differential equation

10dydx

�!� 81�

x2 � 1 " 2 exp � � 5y �has an implicit general solution of the form F � x � y �� K �In fact, because the differential equation is separable, we can



F � x � y �� G � x � � H � y � �19.(1 pt) The differential equation

dydx

� 45� y1 " 2 � 81x2 y1 " 2 �has an implicit general solution of the form F � x � y �� K �In fact, because the differential equation is separable, we can



F � x � y �� G � x � � H � y � �20.(1 pt) The differential equation� 12

�5cos � x �� dy

dx� sin � x � cos � y �

has an implicit general solution of the form F � x � y �� K �In fact, because the differential equation is separable, we can



F � x � y �� G � x � � H � y � �21.(1 pt) A.Find y in terms of x if

dydx

� x6y 7

and y � 0 � � 7 �y � x ��

B. For what x-interval is the solution defined?(Your answers should be numbers or plus or minus infinity.For plus infinity enter ”PINF” ; for minus infinity enter”MINF”.)The solution is defined on the interval:� x �


� � 12x�

7 �� 27y2�

8y�

8 �

has an implicit general solution of the form F � x � y �� K �In fact, because the differential equation is separable, we can



F � x � y �� G � x � � H � y � �23.(1 pt) The differential equation

exp � y � dydx

� � 2x�

7 �� 3 sin � y � � 9 cos � y ��has an implicit general solution of the form F � x � y �� K �

In fact, because the differential equation is separable, we candefine the solution curve implicitly by a function in the formF � x � y �� G � x � � H � y � � K


F � x � y �� G � x � � H � y � �24.(1 pt) A. Solve the following initial value problem:

cos � t � 2 dydt

� 1

with y � 25 � � tan � 25 ��(Find y as a function of t.)y �

B. On what interval is the solution valid?(Your answer should involve pi.)Answer: It is valid for � t � .

C. Find the limit of the solution as t approaches the left endof the interval.(Your answer should be a number or ”PINF” or ”MINF”.”PINF” stands for plus infinity and ”MINF” stands for minusinfinity.)Answer C:

D. Similar to C, but for the right end.Answer D:


�� 12�

12x�

18y�

18xy �has an implicit general solution of the form F � x � y �� K �In fact, because the differential equation is separable, we can



F � x � y �� G � x � � H � y � �Prepared by the WeBWorK group, Dept. of Mathematics, University of Rochester, c

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dydx

�8y � 7

satisfying the initial condition y � 0 � � 0.Answer: y=Your answer should be a function of x.

2.(1 pt) GUESS one function y � t � which solves the problembelow, by determining the general form the function might takeand then evaluating some coefficients.

6tdydt

�y � t4

Find y � t � .y � t � �

3.(1 pt) GUESS one function y � t � which solves the problembelow, by determining the general form the function might takeand then evaluating some coefficients.

dydt

�3y � exp � 3t �

Find y � t � .y � t � �

4.(1 pt) Find the function satisfying the differential equation

y� �

5y � 2e7t

and y � 0 � � �4.

5.(1 pt) Solve the following initial value problem:

tdydt

�6y � 3t

with y � 1 �� 4 �(Find y as a function of t.)y �

6.(1 pt) Solve the following initial value problem:dydt

� � 0 � 2 � ty � 6t


7.(1 pt) Solve the initial value problem

9 � t � 1 � dydt

�5y � 20t �

for t � �1 with y � 0 � � 16 �

y �8.(1 pt) Find the particular solution of the differential equa-

tiondydx

�ycos � x �� 2cos � x �

satisfying the initial condition y � 0 � � 4.Answer: y=Your answer should be a function of x.


dydt

�y � 5exp � t � � 3exp � 4t �

with y � 0 �� 4 �y �

10.(1 pt) Solve the initial value problemdydt

�2y � 30sin � t � � 15cos � t �

with y � 0 �� 7 �y �

11.(1 pt) Solve the following initial value problem:

7dydt

�y � 56t



9 � sin � t � dydt

� � cos t � y � �!� cos � t ��#� sin � t �� 4 �for 0 � t � π and y � π $ 2 � � 13 �y �

13.(1 pt) A. Let g(t) be the solution of the initial value prob-lem

4tdydt

�y � 0 � t � 0 �

with g � 1 � � 1 �Find g � t � .g � t � �

B. Let f � t � be the solution of the initial value problem

4tdydt

�y � t5

with f � 0 �� 0 �Find f � t � .f � t ��(Hint: you can try to guess this solution.)

C. Find a constant c so that

k � t �� f � t � � cg � t �solves the differential equation in part B and k(1) = 8.c �

14.(1 pt) A. Let g � t � be the solution of the initial value prob-lem

dydt

�5y � 0 �

with y � 0 �� 1 �Find g � t � .g � t � �

B. Let f � t � be the solution of the initial value problemdydt

�5y � exp � 5t �

with y � 0 �� 1 $ 10 �Find f � t � .f � t ��

1

C. Find a constant c so that

k � t �� f � t � � cg � t �solves the differential equation in part B and k(0) = 18.c �

15.(1 pt) Find a family of solutions to the differential equa-tion � x2 � 1xy � dx

�xdy � 0

(To enter the answer in the form below you may have to re-arrange the equation so that the constant is by itself on one sideof the equation.) Then the solution in implicit form is:

the set of points (x, y) where F(x,y) =

= constant


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1.(1 pt) A tank contains 2180 L of pure water.A solution thatcontains 0.07 kg of sugar per liter enters a tank at the rate 3L/min The solution is mixed and drains from the tank at thesame rate.(a) How much sugar is in the tank initially?

(kg)(b) Find the amount of sugar in the tank after t minutes.amount =

(function of t)(c) Find the concentration of sugar in the solution in the tank

after 81 minutes.concentration =

2.(1 pt) A tank contains 1960 L of pure water. A solution thatcontains 0.07 kg of sugar per liter enters tank at the rate 5 L/min.The solution is mixed and drains from the tank at the same rate.

(a) How much sugar is in the tank at the beginning.y � 0 � � (include units)

(b) With S representing the amount of sugar (in kg) at timet (in minutes) write a differential equation which models thissituation.S� � f � t � S � �

.Note: Make sure you use a capital S, ( and don’t use S(t), itconfuses the computer). Don’t enter units for this function.

(c) Find the amount of sugar (in kg) after t minutes.S � t � � (function of t)(d) Find the amout of the sugar after 60 minutes.S � 60 � � (include units)Click here for help with units

3.(1 pt) A tank contains 2900 L of pure water. Solution thatcontains 0.06 kg of sugar per liter enters the tank at the rate 4L/min, and is thoroughly mixed into it. The new solution drainsout of the tank at the same rate.(a) How much sugar is in the tank at the begining? y � 0 � �(kg)

(b) Find the amount of sugar after t minutes.y � t � � (kg)

(Note that this is a function of t)(c) As t becomes large, what value is y � t � approaching ? In otherwords, calculate

limt % ∞

y � t �(kg)

4.(1 pt) A tank contains 80 kg of salt and 2000 L of water.Asolution of a concentration 0.02 kg of salt per liter enters a tankat the rate 6 L/min. The solution is mixed and drains from thetank at the same rate.

(a) What is the concentration of our solution in the tank ini-tially?

concentration = (kg/L)(b) Find the amount of salt in the tank after 4 hours.amount = (kg)(c) Find the concentration of salt in the solution in the tank

as time approaches infinity.concentration = (kg/L)5.(1 pt) A tank contains 60 kg of salt and 1000 L of water.

Pure water enters a tank at the rate 10 L/min. The solution ismixed and drains from the tank at the rate 5 L/min.

(a) What is the amount of salt in the tank initially?amount = (kg)(b) Find the amount of salt in the tank after 5 hours.amount = (kg)(c) Find the concentration of salt in the solution in the tank

as time approaches infinity. (Assume your tank is large enoughto hold all the solution.)

concentration = (kg/L)6.(1 pt) A tank contains 1640 L of pure water. A solution that

contains 0.03 kg of sugar per liter enters tank at the rate 6 L/minThe solution is mixed and drains from the tank at the same rate.

(a) How much sugar is in the tank at the beginning.y � 0 � � (include units)

(b) Find the amount of sugar (in kg) after t minutes.y � t � � (function of t)(b) Find the amout of the sugar after 84 minutes.y � 84 � � (include units)

7.(1 pt) A cell of some bacteria divides into two cells every50 minutes.The initial population is 4 bacteria.

(a) Find the size of the population after t hoursy � t � �

(function of t)(b) Find the size of the population after 5 hours.y � 5 � �(c) When will the population reach 20?T �8.(1 pt) A cell of some bacteria divides into two cells every

30 minutes. The initial population is 500 bacteria.(a) Find the population after t hoursy � t � � (function of t)(b) Find the population after 7 hours.y � 7 � �(c) When will the population reach 3500?T �9.(1 pt) A bacteria culture starts with 140 bacteria and grows

at a rate proportional to its size. After 4 hours there will be 560bacteria.

(a) Express the population after t hours as a function of t.population: (function of t)(b) What will be the population after 9 hours?

(c) How long will it take for the population to reach 2670 ?1

10.(1 pt) A population P obeys the logistic model. It satisfiesthe equationdPdt � 1

1300 P � 13�

P � for P � 0 �(a) The population is increasing when � P �(b) The population is decreasing when P �(c) Assume that P � 0 � � 3 � Find P � 73 ��11.(1 pt) An unknown radioactive element decays into non-

radioactive substances. In 320 days the radioactivity of a sampledecreases by 31 percent.

(a) What is the half-life of the element?half-life: (days)(b) How long will it take for a sample of 100mg to decay to

59 mg?time needed: (days)

12.(1 pt) A body of mass 3 kg is projected vertically upwardwith an initial velocity 42 meters per second.

The gravitational constant is g � 9 � 8m $ s2. The air resistanceis equal to k & v & where k is a constant.

Find a formula for the velocity at any time ( in terms of k ):v � t � �Find the limit of this velocity for a fixed time t0 as the air

resistance coefficient k goes to 0. (Enter t0 as t 0 .)v � t0 � �How does this compare with the solution to the equation for

velocity when there is no air resistance?This illustrates an important fact, related to the fundamental

theorem of ODE and called ’continuous dependence’ on param-eters and initial conditions. What this means is that, for a fixedtime, changing the initial conditions slightly, or changing theparameters slightly, only slightly changes the value at time t.

The fact that the terminal time t under consideration is afixed, finite number is important. If you consider ’infinite’ t,or the ’final’ result you may get very different answers. Con-sider for example a solution to y’=y, whose initial condition isessentially zero, but which might vary a bit positive or negative.If the initial condition is positive the ”final” result is plus infin-ity, but if the initial condition is negative the final condition isnegative infinity.

13.(1 pt) You have 800 dollars in your bank account. Sup-pose your money is compounded every month at a rate of 0.4percent per month.

(a) How much do you have after t years.y � t � � (function of t)(b) How much do you have after 90 months.y � 90 � �14.(1 pt) A young person with no initial capital invests k dol-

lars per year in a retirement account at an annual rate of return0.06. Assume that investments are made continuously and thatthe return is compounded continuously.

Determine a formula for the sum S � t � – (this will involve theparameter k):

S � t � =What value of k will provide 2753000 dollars in 44 years?

k �15.(1 pt) Here is a somewhat realistic example which com-

bines the work on earlier problems. You should use the phaseplane plotter to look at some solutions graphically before youstart solving this problem and to compare with your analytic an-swers to help you find errors. You will probably be surprisedto find how long it takes to get all of the details of solution of arealistic problem right, even when you know how to do each ofthe steps. There is partial credit on this problem.

There are 1180 dollars in the bank account at the beginningof January 1990, and money is added and withdrawn from theaccount at a rate which follows a sinusoidal pattern, peaking inJanuary and in July with money being added at a rate corre-sponding to 1480 dollars per year, while maximum withdrawalstake place at the rate of 1200 dollars per year in April and Octo-ber.

The interest rate remains constant at the rate of 1 percent peryear, compounded continuously.

Let y � t � represents the amount of money at time t (t is inyears).

y � 0 � � (dollars)Write a formula for the rate of deposits and withdrawals (us-

ing the functions sin(), cos() and constants):g � t � =

The interest rate remains constant at 1 percent per year overthis period of time.With y representing the amount of money in dollars at time t (inyears) write a differential equation which models this situation.y� � f � t � y ��

.Note: Use y rather than y � t � since the latter confuses the com-puter. Don’t enter units for this equation.

Find an equation for the amount of money in the account attime t where t is the number of years since January 1990.y � t � �

(c) Find the amount of money in the bank at thebeginning of January 2000 (10 years later):

Find a solution to the equation which does not become infi-nite (either positive or negative) over time:y � t � �

During which months of the year does this non-growing so-lution have the highest values? ?

16.(1 pt) Newton’s law of cooling states that the tempera-ture of an object changes at a rate proportional to the differencebetween its temperature and that of its surroundings. Supposethat the temperature of a cup of coffee obeys Newton’s law ofcooling. If the coffee has a temperature of 205 degrees Fahren-heit when freshly poured, and 1 minutes later has cooled to 193degrees in a room at 60 degrees, determine when the coffeereaches a temperature of 163 degrees.

2

The coffee will reach a temperature of 163 degrees inminutes.

17.(1 pt) Susan finds an alien artifact in the desert, wherethere are temperature variations from a low in the 30s at night toa high in the 100s in the day. She is interested in how the artifactwill respond to faster variations in temperature, so she kidnapsthe artifact, takes it back to her lab (hotly pursued by the mili-tary police who patrol Area 51), and sticks it in an ”oven” – thatis, a closed box whose temperature she can control precisely.

Let T � t � be the temperature of the artifact. Newton’s lawof cooling says that T � t � changes at a rate proportional to thedifference between the temperature of the environment and thetemperature of the artifact. This says that there is a constantk, not dependent on time, such that T

� � k � E �T � , where E is

the temperature of the environment (the oven). Before collect-ing the artifact from the desert, Susan measured its temperatureat a couple of times, and she has determined that for the alienartifact, k � 0 � 9.

Susan preheats her oven to 70 degrees Fahrenheit (she hasstubbornly refused to join the metric world). At time t � 0 theoven is at exactly 70 degrees and is heating up, and the ovenruns through a temperature cycle every 2π minutes, in which itstemperature varies by 25 degrees above and 25 degrees below70 degrees.

Let E � t � be the temperature of the oven after t minutes.E � t � �

At time t � 0, when the artifact is at a temperature of 85 de-grees, she puts it in the oven. Let T � t � be the temperature of theartifact at time t. Then T � 0 �� (degrees)

Write a differential equation which models the temperatureof the artifact.T� � f � t � T � �

.Note: Use T rather than T � t � since the latter confuses the com-puter. Don’t enter units for this equation.

Solve the differential equation. To do this, you may find ithelpful to know that if a is a constant, then'

sin � t � eat dt � 1a2�

1eat � asin � t � � cos � t �� C �

T � t � �After Susan puts in the artifact in the oven, the military po-

lice break in and take her away. Think about what happens toher artifact as t ( ∞ and fill in the following sentence:

For large values of t, even though the oven temperature variesbetween 45 and 95 degrees, the artifact varies from

to

degrees.(To answer, you will need to use techniques you reviewed in thetrig problems on this assignment to assemble two trig functionsinto one.)

18.(1 pt) Here is a multipart example on finance. Be patientand careful as you work on this problem. You will probably besurprised to find how long it takes to get all of the details of so-lution of a realistic problem right, even when you know how todo each of the steps. Use the computer to check the steps foryou as you go along. There is partial credit on this problem.

A recent college graduate borrows 50000 dollars at an (an-nual) interest rate of 9.5 per cent. Anticipating steady salaryincreases, the buyer expects to make payments at a monthly rateof 775 � 1 � t $ 120 � dollars per month, where t is the number ofmonths since the loan was made.

Let y � t � be the amount of money that the graduate owes tmonths after the loan is made.

y � 0 � � (dollars)With y representing the amount of money in dollars at time t

(in months) write a differential equation which models this situ-ation.y� � f � t � y ��

.Note: Use y rather than y � t � since the latter confuses the com-puter. Don’t enter units for this equation.

Find an equation for the amount of money owed after tmonths.y � t � �

Next we are going to think about how many months it willtake until the loan is paid off. Remember that y � t � is the amountthat is owed after t months. The loan is paid off when y � t � =

Once you have calculated how many months it will taketo pay off the loan, give your answer as a decimal, ignoringthe fact that in real life there would be a whole number ofmonths. To do this, you should use a graphing calculator oruse a plotter on this page to estimate the root. If you use thethe xFunctions plotter, then once you have launched xFunc-tions, pull down the Multigaph Utility from the menu in the up-per right hand corner, enter the function you got for y (using xas the independent variable, sorry!), choose appropriate rangesfor the axes, and then eyeball a solution.

The loan will be paid off in months.If the borrower wanted the loan to be paid off in exactly 20

years, with the same payment plan as above, how much couldbe borrowed?Borrowed amount =


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3


1.(1 pt) The graph of the function f � x � is

(the hor-izontal axis is x.)

Given the differential equation x� � t � � f � x � t �� .

List the constant (or equilibrium) solutions to this differentialequation in increasing order and indicate whether or not theseequations are stable, semi-stable, or unstable.

?

?

?

?

2.(1 pt) The graph of the function f � x � is

(the hor-izontal axis is x.)

Given the differential equation x� � t � � f � x � t �� .

List the constant (or equilibrium) solutions to this differentialequation in increasing order and indicate whether or not theseequations are stable, semi-stable, or unstable.

?

?

?

?

3.(1 pt) Given the differential equation x� �)� x � 3 �+*,� x �

1 � 3 � x � 1 � 2 � x � 2 � 5 � .List the constant (or equilibrium) solutions to this differen-

tial equation in increasing order and indicate whether or notthese equations are stable, semi-stable, or unstable. (It helpsto sketch the graph. xFunctions will plot functions as well asphase planes. )

?

?

?

?

4.(1 pt) Given the differential equation x� � t �-� x4

�1x3

�13x2

�1x�

12.List the constant (or equilibrium) solutions to this differen-

tial equation in increasing order and indicate whether or notthese equations are stable, semi-stable, or unstable. (It helpsto sketch the graph. xFunctions will plot functions as well asphase planes. )

?

?

?

?


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1.(1 pt) The following differential equation is exact.Find a function F(x,y) whose level curves are solutions to the

differential equation

ydy�

xdx � 0

F(x,y) =

2.(1 pt) Use the ”mixed partials” check to see if the followingdifferential equation is exact.If it is exact find a function F(x,y) whose level curves are solu-tions to the differential equation� 1x3 � 4y � dx

� � 3x�

2y1 � dy � 0

?F(x,y) =

3.(1 pt) Use the ”mixed partials” check to see if the followingdifferential equation is exact.If it is exact find a function F(x,y) whose level curves are solu-tions to the differential equation� 1xy2 � 2y � dx

� � 1x2y�

2x � dy � 0

?F(x,y) =

4.(1 pt) Use the ”mixed partials” check to see if the followingdifferential equation is exact.

If it is exact find a function F(x,y) whose level curves aresolutions to the differential equation

dydx

� �3x3 � 4y

0x�

1y3

?F(x,y) =5.(1 pt) Use the ”mixed partials” check to see if the following

differential equation is exact.If it is exact find a function F(x,y) whose level curves are solu-tions to the differential equation� 3ex sin � y � � 2y � dx

� � 2x�

3ex cos � y �� dy � 0

?F(x,y) =6.(1 pt) Check that the equation below is not exact but be-

comes exact when multiplied by the integrating factor.

x2y3 � x � 1 � y2 � y � � 0

Integrating factor: µ � x � y �� 1 $.� xy3 � .Solve the differential equation.

You can define the solution curve implicitly by a function in theformF � x � y �� G � x � � H � y � � K F(x,y) =

7.(1 pt) Find an explicit or implicit solutions to the differen-tial equation � x2 � 3xy � dx

�xdy � 0

F(x,y) =


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1.(1 pt) This problem involves using the uniqueness property (from the Fundamental Theorem of ordinary differential equations.)It can’t be graded by WeBWorK, but is to be handed in at the first class after the due date.

A. State the uniqueness property of the fundamental theorem.

B. Show directly using the differential equation, that if y1 � t � is a solution to the differential equation y� � t �/� y � t � , then y2 � t ��

y1 � t � a � is also a solution to the differential equation. (You will need to use the known facts about y1 to calculate that y�2 � t �� y2 � t �

). (We know that the solution is the exponential function, but you will not need to use this fact.)

C. Describe the relationship between the graphs of y1 and y2 and using a sketch of the direction field explain why it is obviousthat if y1 is a solution then y2 has to be a solution also.

D. Describe in words why if y1 � t � is any solution to the differential equation y� � f � y � then y2 � t �� y1 � t � a � is also a solution.

E. Show that if y1 � t � solves y� � t � � y � t � , then y2 � t �� Ay1 � t � also solves the same equation.

F. Suppose that y1 � t � solves y� � t �� y � t � and y � 0 �� 1. (Such a solution is guaranteed by the fundamental theorem.). Let y2 � t ��

y1 � t � a � and let y3 � t � � y1 � a � y1 � t � . Calculate the values y2 � 0 � and y3 � 0 � . Use the uniqueness property to show that y2 � t � � y3 � t � forall t.

G. Explain how this proves that any solution to y� � y must be a function which obeys the law of exponents.

H. Let z � x�

iy. Define exp � z � ( or ez ) using a Taylor series. Show that if z � x�

iy is a constant, then

ddt

exp � tz �� zexp � tz �by differentiating the power series.

I. Use your earlier results to show that exp � z � w �0� exp � z � exp � w � . This method of checking the law of exponents is MUCHeasier than expanding the power series.

You can find a direction field plotter here or at thedirection field plotter page . Choose ”integral curves utility” from the ”main screen” menu of xFunctions to get to the phaseplane

plotter.


A. Using the same technique as in the previous problem show that if a function y1 � t � satisfies: (1) y1 � 0 � = 1 and (2) y� � t �� y � t �

then � y1 � t �� r � y1 � rt �B. Explain in words how this relates to another law of exponents.

You can find a direction field plotter htmlLink(”http://webwork.math.rochester.edu/mth163/phaseplaneplotters/launchXfunctions.html”,”here”) or at the

htmlLink( ”http://webwork.math.rochester.edu/mth163/phaseplaneplotters/”, ”direction field plotter page”). Choose ”integralcurves utility” from the ”main screen” menu of xFunctions to get to the phaseplane plotter.


Use the same ideas as in the previous problems.1

A. Suppose that y1 � t � satisfies the equation y� �1�

y � 0 and y1 � 0 �-� 0and y�1 � 0 �2� 1. Such a function exists because of the

fundamental theorem. (We all know that it is sin � t � , but you should not use that fact in answering the questions below.)Show that y2 � t �� y

�1 � t � also satisfies the equation y

� ��y � 0 and that y2 � 0 � � 1 and y

�2 � 0 � � 0.

B. If y3 � t �� y�2 � t � show, using the uniqueness property, that y3 � t � � �

y1 � t �C. State the uniqueness property for solutions to second order differential equations (or equivalently to a system of two first order

differential equations).

D. Use the uniqueness property to show that y1 � t � a � � y�1 � a � y1 � t � � y1 � a � y2 � t �� y2 � a � y1 � t � � y1 � a � y2 � t �

The formulas for the sin of sums of angles can be calculated completely from the one fact that it satisfies a differential equation.This is a general fact. Any solution of a differential equation has the potential for obeying certain ”laws” which are dictated by thedifferential equation.


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1.(1 pt) Find y as a function of t if10000y

� � �9y � 0 �

y � 0 � � 3 � y� � 0 � � 4 �

y � t � �2.(1 pt) Find y as a function of t if

2500y� � �

81y � 0

with y � 0 �� 3 � y� � 0 �� 4 �

y �3.(1 pt) Find y as a function of t if

16y� ��

81y � 0 � y � 0 �� 4 � y� � 0 �� 1 �


y� � �

10y� �

24y � 0

y � 0 � � 6 � y � 1 � � 7 �y � t � �Remark: The initial conditions involve values at two points.


� ��96y

��32y � 0 � y � 0 � � 7 � y

� � 0 � � 2 �y �

6.(1 pt) Find y as a function of t if4y� � �

30y � 0

y � 0 � � 8 � y� � 0 � � 4 �

y � t � �Note: This particular weBWorK problem can’t handle com-

plex numbers, so write your answer in terms of sines andcosines, rather than using e to a complex power.


� � �28y

� �6y � 0

y � 0 � � 7 � y� � 0 � � 2 �

y �Note: This problem cannot interpret complex numbers. You

may need to simplify your answer before submitting it.


� � �4y� �

16y � 0

y � 0 � � 4 � y� � 0 � � 2 �

y � t � �Note: This problem cannot interpret complex numbers. You


9.(1 pt) Find y as a function of t ify� ��

16y��

113y � 0 � y � 0 �� 7 � y� � 0 �� 6 �

y �Note: This problem cannot interpret complex numbers. You



� ��126y

��32y � 0 � y � 0 �� 6 � y

� � 0 �� 8 �y �

Note: This problem cannot interpret complex numbers. Youmay need to simplify your answer before submitting it.

11.(1 pt) Find the function y1 of t which is the solution of36y

� ��24y

� � 0with initial conditions y1 � 0 �� 1 � y

�1 � 0 �� 0 �

y1 �Find the function y2 of t which is the solution of 36y

� ��24y

� � 0with initial conditions y2 � 0 �� 0 � y

�2 � 0 �� 1 �

y2 �Find the Wronskian W � t �� W � y1

� y2 ��W � t ��Remark: You can find W by direct computation and use Abel’stheorem as a check. You should find that W is not zero and so y1

and y2form a fundamental set of solutions of 36y� ��

24y� � 0 �

12.(1 pt) Find y as a function of t if4y� �#�

68y��

289y � 0 � y � 0 � � 5 � y� � 0 � � 9 �


25y� ��

20y��

4y � 0 � y � 3 �� 6 � y� � 3 �� 1 �

y �14.(1 pt) Determine whether the following pairs of functions

are linearly independent or not.

? 1. f � t �� t and g � t � �!& t &? 2. The Wronskian of two functions is W � t �-� t are the

functions linearly independent or dependent?? 3. f � θ �� 19cos3θ and g � θ �� 76cos3 θ

�57cosθ

15.(1 pt)Suppose that the Wronskian of two functions f1 � t � and f2 � t �

is given by

W � t � � t2 � 4 � det 3 f1 � t � f2 � t �f�1 � t � f

�2 � t �54

Even though you don’t know the functions f1 and f2 you candetermine whether the following questions are true or false.

? 1. The vectors � f1 � � 2 � � f �1 � � 2 �� and � f2 � � 2 � � f �2 � � 2 ��are linearly independent

? 2. The equations

a f1 � 2 � � b f2 � 2 �6� 0a f�1 � 2 � � b f

�2 � 2 �6� 0

have more than one solution.? 3. The equations

a f1 � 2 � � b f2 � 2 �7� ca f�1 � 2 � � b f

�2 � 2 �8� d

have a unique solution for any c and d? 4. The vectors � f1 � 0 � � f �1 � 0 �� and � f2 � 0 � � f �2 � 0 �� are lin-

early independent1

? 5. The equations

a f1 � 0 � � b f2 � 0 �7� ca f�1 � 0 � � b f

�2 � 0 �8� d

have a unique solution for any c and d

16.(1 pt)Determine which of the following pairs of functions are lin-

early independent.

? 1. f � θ �� cos � 3θ � � g � θ �� 2cos3 � θ � � 4cos � θ �? 2. f � t �� 2t2 � 14t � g � t � � 2t2 � 14t

? 3. f � t �� 3t � g � t � �!& t &? 4. f � x �� e2x � g � x �� e2 � x 3 �17.(1 pt)Match the second order linear equations with the Wronskian

of (one of) their fundamental solution sets.

1. y� �� 2

t y��

4y � 02. y

� � � 2t y� �

4y � 0

3. y� ��

1y��

4y � 04. y

� ��1y��

4y � 05. y

� ��2ty

��4y � 0

A. W � t � � e1t

B. W � t � � t2

C. W � t � � 3exp � t2 �D. W � t � � 5

t2

E. W � t � � 2e 1t

18.(1 pt) Find y as a function of x if

x2y� � �

2xy� �

72y � 0 �y � 1 � � �

1 � y� � 1 ��

7 �y �19.(1 pt) Find y as a function of x if

x2y� � �

15xy� �

64y � 0 �y � 1 � � 6 � y

� � 1 � � �5 �

y �


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2


1.(1 pt) Find a single solution of y if y� � � 2 �

y �2.(1 pt) Use the method of undetermined coefficients to find

one solution ofy� � �

8y� �

6y �!� 3 � exp �� 8 �9* t �.

y �(It doesn’t matter which specific solution you find for this

problem.)3.(1 pt) There is an error in this problem — it has been

marked correct for everyone. I’ll get a replacement for it readyfor the next problem set. Sorry about that.

Take care,MikeUse the method of undetermined coefficients to find one so-

lution ofy� ��

3y�#�

3y �!� � 8 t2 � 0 t�

6 � exp � 3 t � .Note that the method finds a specific solution, not the generalone. y �

4.(1 pt) Use the method of undetermined coefficients to findone solution ofy� ��

4y��

17y �48exp � 2t � cos � 3t � � 32exp � 2t � sin � 3t � � 3 * exp �� 3 �9* t � .(It doesn’t matter which specific solution you find for this prob-lem.)y �

5.(1 pt) Use the method of undetermined coefficients to findone solution ofy� � �

2y� �

2y �:� 10t�

7 � exp � � t � cos � t � � � 11t�

25 � exp � � t � sin � t �.(It doesn’t matter which specific solution you find for this prob-lem.)y �

6.(1 pt)Find a particular solution to the differential equation

y� � �

1y� �

20y � �1200t3 �

yp �7.(1 pt)Find a particular solution to y

� �#�6y��

9y � 6 � 5e3t .yp �

8.(1 pt)Find a particular solution to the differential equation

2y� � �

1y� �

1y � �1t2 � 2t

�1e 2t �


� � �5y� �

4y � �3te4t .


� ��16y � �

16sin � 4t � .yp �

11.(1 pt) Find the solution of y� ��

6y��

7y � 60exp �� 3 � t �with y � 0 �� 2 and y

� � 0 �� 3 �y �12.(1 pt) Find the solution of

y� � �

2y� �

y � 324exp �� 8 � t �with y � 0 �� 3 and y

� � 0 �� 6 �y �13.(1 pt) Find the solution of

y� � �

3y� � 36 sin � 3t � � 36 cos � 3t �

with y � 0 �� 1 and y� � 0 �� 6 �

y �14.(1 pt) Find y as a function of x if

x2y� � �

3xy� �

35y � x8 �y � 1 � � �

6 � y� � 1 �� 3 �

y �15.(1 pt) Find y as a function of x if

x2y� � �

15xy� �

49y � x9 �y � 1 � � �

1 � y� � 1 �� 2 �

y �


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1.(1 pt) Another ”realistic” problem:The following problem is similar to the problem in an earlier

assignment about the bank account growing with periodic de-posits. The basic procedure for this problem is not too hard, butgetting details of the calculation correct is NOT easy, and maytake some time.

A ping-pong ball is caught in a vertical plexiglass column inwhich the air flow alternates sinusoidally with a period of 60seconds. The air flow starts with a maximum upward flow atthe rate of 3m $ s and at t � 30 seconds the flow has a minimum(upward) flow of rate of

�5 � 6m $ s. (To make this clear: a flow

of�

5m $ s upward is the same as a flow downward of 5m $ s.The ping-pong ball is subjected to the forces of gravity

(�

mg) where g � 9 � 8m $ s2 and forces due to air resistance whichare equal to k times the apparent velocity of the ball through theair.

What is the average velocity of the air flow? You can aver-age the velocity over one period or over a very long time – theanswer should come out about the same – right?

. (Include units).Write a formula for the velocity of the air flow as a function

of time.A � t � �

Write the differential equation satisfied by the velocity ofthe ping-pong ball (relative to the fixed frame of the plexiglasstube.) The formulas should not have units entered, but use unitsto trouble shoot your answers. Your answer can include the pa-rameters m - the mass of the ball and k the coefficient of airresistance, as well as time t and the velocity of the ball v. (Usejust v, not v(t) the latter confuses the computer.)v� � t � �

Use the method of undetermined coefficients to find one pe-riodic solution to this equation:v � t � =

Find the amplitude and phase shift of this solution. You donot need to enter units.v � t � � cos � * t

� �Find the general solution, by adding on a solution to the ho-

mogeneous equation. Notice that all of these solutions tend to-wards the periodically oscillating solution. This is a general-ization of the notion of stability that we found in autonomousdifferential equations.

Calculate the specific solution that has initial conditions t � 0and w � 0 � � 2 � 4.

w � t ��Think about what effect increasing the mass has on the am-

plitude, on the phase shift? Does this correspond with your ex-pectations?

2.(1 pt) A steel ball weighing 128 pounds is suspended froma spring. This stretches the spring 128

145 feet.The ball is started in motion from the equilibrium position

with a downward velocity of 8 feet per second.The air resistance (in pounds) of the moving ball numericallyequals 4 times its velocity (in feet per second) .

Suppose that after t seconds the ball is y feet below its restposition. Find y in terms of t. (Note that this means that thepostiive direction for y is down.)

y �Take as the gravitational acceleration 32 feet per second per

second.

3.(1 pt) A hollow steel ball weighing 4 pounds is suspendedfrom a spring. This stretches the spring 1

5 feet.The ball is started in motion from the equilibrium position

with a downward velocity of 4 feet per second. The air resis-tance (in pounds) of the moving ball numerically equals 4 timesits velocity (in feet per second) .

Suppose that after t seconds the ball is y feet below its restposition. Find y in terms of t. (Note that the positive directionis down.)

Take as the gravitational acceleration 32 feet per second persecond.

y �4.(1 pt) This problem is an example of critically damped har-

monic motion.A hollow steel ball weighing 4 pounds is suspended from a

spring. This stretches the spring 18 feet.

The ball is started in motion from the equilibrium positionwith a downward velocity of 5 feet per second. The air resis-tance (in pounds) of the moving ball numerically equals 4 timesits velocity (in feet per second) . Suppose that after t secondsthe ball is y feet below its rest position. Find y in terms of t.

Take as the gravitational acceleration 32 feet per second persecond. (Note that the positive y direction is down in this prob-lem.)

y �1


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1.(1 pt) Match the third order linear equations with their fun-damental solution sets.

1. y� � ��

y� � 0

2. ty� � �#�

y� � � 0

3. y� � ��

7y� ��

10y� � 0

4. y� � � �

6y� � �

y� �

6y � 05. y

� � ��3y� ��

3y��

y � 06. y

� � ��y� �#�

y��

y � 0

A. et � tet � e t

B. e t � te t � t2e t

C. 1 � cos � t � � sin � t �D. e6t � cos � t � � sin � t �E. 1 � e5t � e2t

F. 1 � t � t3

2.(1 pt) Find y as a function of x ify� � � �

12y� � �

35y� � 0 �

y � 0 � � 9 � y� � 0 �� 7 � y

� � � 0 � � 2 �y � x ��


64y� � 0 �

y � 0 � � �1 � y

� � 0 � � 56 � y� � � 0 � � 64 �

y � x ��4.(1 pt) Find y as a function of x if

y � 4 � � 6y� � � �

9y� � � 0 �

y � 0 � � 7 � y� � 0 �� 4 � y

� � � 0 � � 9 � y� � � � 0 � � 0 �

y � x ��5.(1 pt) Find y as a function of x if

y� � � �

5y� � �

y� �

5y � 0 �y � 0 � � 7 � y

� � 0 �� 9 � y

� � � 0 �� 55 �y � x ��

6.(1 pt) If L � D2�

2xD�

3x and y � x �� 2x�

2e2x � thenLy �


16y� � �

63y� � 240ex �

y � 0 � � 14 � y� � 0 � � 26 � y

� � � 0 � � 26 �y � x ��

8.(1 pt) Find y as a function of x ify � 4 � � 12y

� � � �36y

� � � �256e 2x �

y � 0 � � 18 � y� � 0 � � 17 � y

� � � 0 � � 32 � y� � � � 0 � � 8 �

y � x ��


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1.(1 pt) Write the given second order equation as its equiva-lent system of first order equations.

u� � �

1u� �

6u � 0

Use v to represent the ”velocity function”, i.e. v � u� � t � .

Use v and u for the two functions, rather than u � t � and v � t � . (Thelatter confuses webwork. Functions like sin � t � are ok.)

u� �

v� �

Now write the system using matrices:

ddt

3 uv 4 � 3 4 3 u

v 42.(1 pt) Write the given second order equation as its equiva-

lent system of first order equations.

u� � �

7 � 5u� �

7 � 5u � �5sin � 3t � u � 1 � � 7 � u � � 1 � � 8 � 5



u� �

v� �


ddt

3 uv 4 � 3 4 3 u

v 4 � 3 4and the initial value for the vector valued function is:3 u � 1 �

v � 1 � 4 � 3 43.(1 pt) Write the given second order equation as its equiva-

lent system of first order equations.

t2u� � �

1 � 5tu� � � t2 � 5 � u � 1 � 5sin � 3t �



u� �

v� �


ddt

3 uv 4 � 3 4 3 u

v 4 � 3 4

4.(1 pt) This problem is similar to problem 21 on page 346.Consult that page for the diagram. You will probably want towrite the solution out first, before trying to enter the answersinto the computer.

Consider two interconnected tanks as shown in Fig 7.1.6 onpage 347. Tank 1 initial contains 90 L (liters) of water and 425g of salt, while tank 2 initially contains 80 L of water and 210 gof salt. Water containing 35 g/L of salt is poured into tank1 ata rate of 1 L/min while the mixture flowing into tank 2 containsa salt concentration of 45 g/L of salt and is flowing at the rateof 2.5 L/min. The two connecting tubes have a flow rate of 4L/min from tank 1 to tank 2; and of 3 L/min from tank 2 back totank 1. Tank 2 is drained at the rate of 3.5 L/min.

You may assume that the solutions in each tank are thor-oughly mixed so that the concentration of the mixture leavingany tank along any of the tubes has the same concentration ofsalt as the tank as a whole. (This is not completely realistic, butas in real physics, we are going to work with the approximate,rather than exact description. The ’real’ equations of physics areoften too complicated to even write down precisely, much lesssolve.)

How does the water in each tank change over time?Let p � t � and q � t � be the amount of salt in g at time t in tanks

1 and 2 respectively. Write differential equations forp and q. (As usual, use the symbols p and q rather than p(t) and q(t). )

p� �

q� �

Give the initial values:3 p � 0 �q � 0 � 4 � 3 4

Show the equation that needs to be solved to find a constantsolution to the differential equation:3 4 � 3 4 3 p

q 4A constant solution is obtained if p � t �� for all time t

and q � t � � for all time t.

5.(1 pt)

Match the differential equations and their matrix function solutions:It’s good practice to multiply at least one matrix solution out fully, to make sure that you know how to do it, but you can get the otheranswers quickly by process of elimination and just multiply out one row or one column.

1

1.

y� � t ��<;= �

32 49�

23�64 78

�34

64�

78 34

>?y � t �

2.

y� � t ��@;= �

13�

2 3�15

�18 5�

33�

18 7

>?y � t �

3.

y� � t � �@;= �

86 218�

16073

�49 80

111�

138 165

>?y � t �

A.

y � t � � ;= 1e60t�

2e45t 4e 75t�3e60t 1e45t

�2e 75t�

5e60t 3e45t�

3e 75t

>?B.

y � t � � ;= �1e0t 3e64t

�3e16t�

3e0t 4e64t�

2e16t�5e0t

�4e64t 2e16t

>?C.

y � t � �<;= �1e 4t 2e 8t

�1e 12t

0�

2e 8t 5e 12t�3e 4t 2e 8t 3e 12t

>?6.(1 pt)

Match the differential equations and their vector valued function solutions:It will be good practice to multiply at least one solution out fully, to make sure that you know how to do it, but you can get the otheranswers quickly by process of elimination and just multiply out one row element.

1.

y� � t ��<;= �

32 49�

23�64 78

�34

64�

78 34

>?y � t �

2.

y� � t � �@;= 14 0

�4

2 13�

8�3 0 25

>?y � t �

3.

y� � t � � ;= 15 0 0

4 20�

154 30

�25

>?y � t �

A.

y � t ��<;= 451

>?e13t

B.

y � t � �<;= 011

>?e5t

C.

y � t ��@;= �1�3�5

>?e0t

7.(1 pt) Calculate the eigenvalues of this matrix:[Note– you’ll probably want to use a graphing calculator to

estimate the roots of the polynomial which defines the eigenval-ues. You can use the web version at xFunctions

If you select the ”integral curves utility” from the main menu,will also be able to plot the integral curves of the associated dif-fential equations. ]

A � 3 �15 1 � 4210854715202e

�14

0�

15 4smaller eigenvalue �associated eigenvector � ( , )larger eigenvalue �associated, eigenvector � ( , )

If y� � Ay is a differential equation, how would the solution

curves behave?� A. The solution curves would race towards zero andthen veer away towards infinity. (Saddle)

� B. All of the solution curves would run away from 0.(Unstable node)� C. All of the solutions curves would converge towards0. (Stable node)� D. The solution curves converge to different points

8.(1 pt) Calculate the eigenvalues of this matrix:[Note– you’ll probably want to use a graphing calculator to

estimate the roots of the polynomial which defines the eigenval-ues. You can use the web version at xFunctions

If you select the ”integral curves utility” from the main menu,will also be able to plot the integral curves of the associated dif-fential equations. ]

A � 3 �18 0�36 0 4

smaller eigenvalue �associated eigenvector � ( , )larger eigenvalue �associated, eigenvector � ( , )

2

If y� � Ay is a differential equation, how would the solution

curves behave?� A. The solution curves would race towards zero andthen veer away towards infinity. (Saddle)

� B. All of the solution curves would run away from 0.(Unstable node)� C. The solution curves converge to different points� D. All of the solutions curves would converge towards0. (Stable node)


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