Arnie Pizer Rochester Problem Library Fall...

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ1 due 01/01/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 1.pg

Match each of the following differential equations with a solu-tion from the list below.

1. 2x2y′′+3xy′ = y2. y′′+6y′+8y = 03. y′′+y = 04. y′′−6y′+8y = 0

A. y =1x

B. y = e4x

C. y = cos(x)D. y = e−2x

2. (1 pt) rochesterLibrary/setDiffEQ1/e7 1 1a.pg

Match each of the differential equation with its solution.

1. xy′−y = x2

2. y′′+y = 03. 2x2y′′+3xy′ = y4. y′′+9y′+18y = 0

A. y = sin(x)B. y = e−6x

C. y = x12

D. y = 3x+x2


Match each differential equation to a function which is a solu-tion.FUNCTIONSA. y = 3x+x2,B. y = e−5x,C. y = sin(x),D. y = x

12 ,

E. y = 3exp(8x),DIFFERENTIAL EQUATIONS

1. 2x2y′′+3xy′ = y2. y′ = 8y3. y′′+y = 04. xy′−y = x2

4. (1 pt) rochesterLibrary/setDiffEQ1/osu de 1 3.pg

Match the following differential equations with their solutions.The symbolsA, B, C in the solutions stand for arbitrary con-stants.You must get all of the answers correct to receive credit.

1.d2ydx2 +25y = 0

2.dydx

=−2xy

x2−5y2

3.d2ydx2 +14

dydx

+49y = 0

4.dydx

= 10xy

5.dydx

+21x2y = 21x2

A. y = Ae−7x +Bxe−7x

B. y = Acos(5x)+Bsin(5x)C. 3yx2−5y3 = CD. y = Ae5x2

E. y = Ce−7x3+1

5. (1 pt) rochesterLibrary/setDiffEQ1/ur de 1 2.pg

Just as there are simultaneous algebraic equations (where a pairof 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 will takesome time to make all of the calculations.

y′1 = y1−2y2 y′2 = 3y1−4y2

• A. y1 = ex y2 = ex

• B. y1 = 2e−2x y2 = 3e−2x

• C. y1 = e−x y2 = e−x

• D. y1 = cos(x) y2 =−sin(x)• E. y1 = sin(x) y2 = cos(x)• F. y1 = sin(x)+cos(x) y2 = cos(x)−sin(x)• G. y1 = e4x y2 = e4x

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.


It can be helpful to classify a differential equation, so that wecan predict the techniques that might help us to find a functionwhich solves the equation. Two classifications are theorderof the equation – (what is the highest number of derivativesinvolved) and whether or not the equation islinear .Linearity is important because the structure of the the family ofsolutions to a linear equation is fairly simple. Linear equationscan usually be solved completely and explicitly.

Determine whether or not each equation is linear:

? 1. t2 d2ydt2

+ tdydt

+2y = sint

? 2.d4ydt4

+d3ydt3

+d2ydt2

+dydt

= 1

? 3.d3ydt3

+ tdydt

+(cos2(t))y = t3

? 4.dydt

+ ty2 = 0

1


It is easy to check that for any value of c, the function

y = x2 +cx2

is solution of equation

xy′+2y = 4x2, (x > 0).

Find the value ofc for which the solution satisfies the initialconditiony(9) = 8.c =


The functionsy = x2 +

cx2

are all solutions of equation:

xy′+2y = 4x2, (x > 0).

Find the constantc which produces a solution which also satis-fies the initial conditiony(10) = 7.c =


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 the value ofc for which the solution satisfies the initialconditiony(−5) = 9.c =


The family of functions

y = ce−2x +e−x

is solution of the equation

y′+2y = e−x.

Find the constantc which defines the solution which also satis-fies the initial conditiony(2) = 10.c =


Find the two values ofk for which

y(x) = ekx

is a solution of the differential equationy′′−1y′+0y = 0.

smaller value =larger value =


Find all values ofk for which the functiony = sin(kt) satisfiesthe differential equationy′′+14y= 0. Separate your answers bycommas.


Find the value ofk for which the constant functionx(t) = k is a

solution of the differential equation 7t5 dxdt

+9x−5 = 0.


Which of the following functions are solutions of the differentialequationy′′−14y′+49y = 0?

• A. y(x) = x2e7x

• B. y(x) = xe7x

• C. y(x) = 7xe7x

• D. y(x) = 0• E. y(x) = e−7x

• F. y(x) = e7x

• G. y(x) = 7x

15. (1 pt) rochesterLibrary/setDiffEQ1/dp7 1 1.pg

Consider the curves in the first quadrant that have equations

y = Aexp(6x),

whereA is a positive constant.Different values ofA give different curves. The curves form afamily, F .Let P = (6,1). Let C be the member of the familyF that goesthrough P.A. Let y = f (x) be the equation ofC. Find f (x).f (x) =B. Find the slope atP of the tangent toC.slope =C. A curveD is perpendicular toC atP. What is the slope of thetangent toD at the pointP? slope =D. Give a formulag(y) for the slope at(x,y) of the member ofF that goes through(x,y). The formula should not involveA orx.g(y) =E. A curve which at each of its points is perpendicular to themember of the familyF that goes through that point is calledan orthogonal trajectory toF . Each orthogonal trajectory toFsatisfies the differential equation

dydx

=− 1g(y)

,

whereg(y) is the answer to part D.Find a functionh(y) such thatx = h(y) is the equation of theorthogonal trajectory toF that passes through the pointP.h(y) =

16. (1 pt) rochesterLibrary/setDiffEQ1/dp7 1 2.pg

The solution of a certain differential equation is of the form

y(t) = aexp(5t)+bexp(9t),

wherea andb are constants.The solution has initial conditionsy(0) = 1 andy′(0) = 1.Find the solution by using the initial conditions to get linearequations fora andb.y(t) =

2

Generated by the WeBWorK systemc©WeBWorK Team, Department of Mathematics, University of Rochester

3

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ10Linear2ndOrderNonhom due 01/10/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ10Linear2ndOrderNonhom-

/ur de 10 4.pg

Find a single solution ofy if y′′ = 2.y =


/ur de 10 5.pg

Use the method of undetermined coefficients to find one solu-tion of

y′′−10y′+28y = 6e7t .

y =(It doesn’t matter which specific solution you find for this prob-lem.)


/ur de 10 13.pg

Use the method of undetermined coefficients to find one solu-tion ofy′′+4y′−4y = (6t2 +9t−1)e3t .Note that the method finds a specific solution, not the generalone.y =


/ur de 10 6.pg


y′′−12y′+86y = 48e6t cos(7t)+48e6t sin(7t)+1e3t .

(It doesn’t matter which specific solution you find for this prob-lem.)y =


/ur de 10 7.pg


y′′+2y′+2y = (10t +7)e−t cos(t)+(11t +25)e−t sin(t).(It doesn’t matter which specific solution you find for this prob-lem.)y =


/ur de 10 11.pg

Find a particular solution to the differential equation

y′′−5y′+4y =−48t3.

yp =


/ur de 10 10.pg

Find a particular solution to

y′′+8y′+16y = 3.5e−4t .

yp =


/ur de 10 8.pg

Find a particular solution to the differential equation

−4y′′+0y′+1y =−2t2 +1t +4e4t .

yp =


/ur de 10 12.pg


y′′+5y′+4y = 12te3t .

yp =


/ur de 10 9.pg


y′′+25y =−50sin(5t).

yp =


/ur de 10 1.pg

Find the solution of

y′′+4y′+3y = 6e0t

with y(0) = 2 andy′(0) = 9.y =


/ur de 10 2.pg


y′′+4y′+4y = 54e1t

with y(0) = 5 andy′(0) = 6.y =


/ur de 10 3.pg


y′′+4y′ = 256 sin(4t)+128 cos(4t)

with y(0) = 9 andy′(0) = 5.y =

1


/ur de 10 14.pg

Findy as a function ofx if

x2y′′−3xy′−32y = x5,

y(1) = 7, y′(1) =−7.y =


/ur de 10 15.pg


x2y′′−5xy′+9y = x2,

y(1) = 8, y′(1) =−4.y =


/ur de 10 16.pg


y′′−4y′+4y =6.5e2t

t2 +1.yp =


/ur de 10 17.pg

Find a particular solution toy′′+16y =−8sec(4t).yp =


2

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ11ModelingWith2ndOrder due 01/11/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ11ModelingWith2ndOrder-

/ur de 11 1.pg

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 at therate of 1.2m/s and att = 30 seconds the flow has a minimum(upward) flow of rate of−3.2m/s. (To make this clear: a flowof −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) whereg= 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-rametersm - the mass of the ball andk the coefficient of airresistance, as well as timet and the velocity of the ballv. (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 conditionst = 0andw(0) = 2.5.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?


/ur de 11 2.pg

A steel ball weighing 128 pounds is suspended from a spring.This stretches the spring128

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

with a downward velocity of 7 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 persecond.


/ur de 11 3.pg

A hollow steel ball weighing 4 pounds is suspended from aspring. This stretches the spring1


with 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 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 =


/ur de 11 4.pg

This problem is an example of critically damped harmonic mo-tion.

A hollow steel ball weighing 4 pounds is suspended from aspring. This stretches the spring1


with a downward velocity of 7 feet per second. The air resis-tance (in pounds) of the moving ball numerically equals 4 times

1

its 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 =


2

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ12HigherOrder due 01/12/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ12HigherOrder/ur de 12 1.pg

Match the third order linear equations with their fundamentalsolution sets.

1. ty′′′−y′′ = 02. y′′′+3y′′+3y′+y = 03. y′′′+y′ = 04. y′′′−8y′′+y′−8y = 05. y′′′−7y′′+12y′ = 06. y′′′−y′′−y′+y = 0

A. 1, t, t3

B. e8t , cos(t), sin(t)C. et , tet , e−t

D. e−t , te−t , t2e−t

E. 1, cos(t), sin(t)F. 1, e4t , e3t


Findy as a function ofx ify′′′−11y′′+24y′ = 0,

y(0) = 3, y′(0) = 3, y′′(0) = 6.y(x) =


Findy as a function ofx ify′′′+64y′ = 0,

y(0) =−3, y′(0) = 16, y′′(0) =−128.

y(x) =


Findy as a function ofx ify(4)−6y′′′+9y′′ = 0,

y(0) = 13, y′(0) = 4, y′′(0) = 9, y′′′(0) = 0.y(x) =


Findy as a function ofx ify′′′−5y′′−y′+5y = 0,

y(0) =−9, y′(0) = 7, y′′(0) = 87.y(x) =


If L = D2 +3xD−2x andy(x) = 2x−5e4x, thenLy =


Findy as a function ofx ify′′′−8y′′+15y′ = 8ex,

y(0) = 14, y′(0) = 28, y′′(0) = 29.y(x) =


Findy as a function ofx ify(4)−10y′′′+25y′′ =−196e−2x,

y(0) = 12, y′(0) = 13, y′′(0) = 21, y′′′(0) = 8.y(x) =


1

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ13Systems1stOrder due 01/13/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder/urde 13 1.pg

Write the given second order equation as its equivalent systemof first order equations.

u′′+3u′+3u = 0

Usev to represent the ”velocity function”, i.e.v = u′(t).Usev andu for the two functions, rather thanu(t) andv(t). (Thelatter confuses webwork. Functions like sin(t) are ok.)u′ =v′ =Now write the system using matrices:ddt

[uv

]=

[ ] [uv

].



u′′+2.5u′+2u = 6sin(3t), u(1) =−5.5, u′(1) =−4

Usev to represent the ”velocity function”, i.e.v = u′(t).Usev andu for the two functions, rather thanu(t) andv(t). (Thelatter confuses webwork. Functions like sin(t) are ok.)u′ =v′ =Now write the system using matrices:ddt

[uv

]=

[ ] [uv

]+

[ ]and the initial value for the vector valued function is:[

u(1)v(1)

]=

[ ].



t2u′′+2.5tu′+(t2 +7)u = 4.5sin(3t)

Use v to represent the ”velocity function”, i.e.v = u′(t).Usev andu for the two functions, rather thanu(t) andv(t). (Thelatter confuses webwork. Functions like sin(t) are ok.)u′ =v′ =Now write the system using matrices:ddt

[uv

]=

[ ] [uv

]+

[ ].


Consider two interconnected tanks as shown in the figureabove. Tank 1 initial contains 100 L (liters) of water and 435g of salt, while tank 2 initially contains 60 L of water and 395 gof salt. Water containing 45 g/L of salt is poured into tank1 at arate of 2 L/min while the mixture flowing into tank 2 contains asalt concentration of 25 g/L of salt and is flowing at the rate of 4L/min. The two connecting tubes have a flow rate of 5.5 L/minfrom tank 1 to tank 2; and of 3.5 L/min from tank 2 back to tank1. Tank 2 is drained at the rate of 6 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) andq(t) be the amount of salt in g at time t in tanks

1 and 2 respectively. Write differential equations forp andq.(As usual, use the symbolsp andq rather thanp(t) andq(t).)p′ =

q′ =

Give the initial values:[p(0)q(0)

]=

[ ].

Show the equation that needs to be solved to find a constantsolution to the differential equation:[ ]

=[ ] [

pq

].

A constant solution is obtained ifp(t) = for all time tandq(t) = for all time t.


Match the differential equations and their matrix function solu-tions:

1

It’s good practice to multiply at least one matrix solution outfully, to make sure that you know how to do it, but you canget the other answers quickly by process of elimination and justmultiply out one row or one column.

1. y′(t) =

-86 218 -16073 -49 80111 -138 165

y(t)

2. y′(t) =

15 0 04 20 -154 30 -25

y(t)

3. y′(t) =

-2 0 00 -2 00 0 -2

y(t)

A. y(t) =

−5e−2t 2e−2t 3e−2t

−2e−2t 2e−2t 2e−2t

e−2t 0 4e−2t

B. y(t) =

5e15t 0 02e15t e5t 2e10t

2e15t e5t 4e10t

C. y(t) =

e60t −2e45t 4e−75t

−3e60t e45t −2e−75t

−5e60t 3e45t −3e−75t


Match the differential equations and their vector valued functionsolutions: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 theother answers quickly by process of elimination and just multi-ply out one row element.

1. y′(t) =

14 0 -42 13 -8-3 0 25

y(t)

2. y′(t) =

-97 33 -5-140 84 35-4 15 -8

y(t)

3. y′(t) =

-86 218 -16073 -49 80111 -138 165

y(t)

A. y(t) =

-2-44

e−21t

B. y(t) =

-213

e45t

C. y(t) =

451

e13t


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 atxFunctions.

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 =[

-20 00 0

]smaller eigenvalue=associated eigenvector= ( , )larger eigenvalue=associated, eigenvector= ( , )

If y′ = Ay is a differential equation, how would the solutioncurves behave?

• A. All of the solutions curves would converge towards0. (Stable node)

• B. The solution curves converge to different points.• C. The solution curves would race towards zero and

then veer away towards infinity. (Saddle)• D. All of the solution curves would run away from 0.

(Unstable node)


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 atxFunctions.

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 =[

-60 -7056 108

]smaller eigenvalue=associated eigenvector= ( , )larger eigenvalue=associated, eigenvector= ( , )

If y′ = Ay is a differential equation, how would the solutioncurves behave?

• A. All of the solutions curves would converge towards0. (Stable node)

• B. All of the solution curves would run away from 0.(Unstable node)

• C. The solution curves would race towards zero andthen veer away towards infinity. (Saddle)

• D. The solution curves converge to different points.

2


Consider the following model for the populations of rabbits andwolves (whereR is the population of rabbits andW is the popu-lation of wolves).

dRdt

= 0.16R(1−0.0004R)−0.002RW

dWdt

= −0.04W+8e−05RW

Find all the equilibrium solutions:(a) In the absence of wolves, the population of rabbits ap-proaches .(b) In the absence of rabbits, the population of wolves ap-proaches .(c) If both wolves and rabbits are present, their populations ap-proachr = andw = .

10. (1 pt) rochesterLibrary/setDiffEQ13Systems1stOrder-

/ur de 13 10.pg

Solve the systemdxdt

=[

-12 5-30 13

]x

with the initial valuex(0) =[

821

].

x(t) =[ ]

.


/ur de 13 11.pg


=[

-12 -12-9 -9

]x

with the initial valuex(0) =[

1315

].

x(t) =[ ]

.


/ur de 13 12.pg

Consider the interaction of two species of animals in a habitat.We are told that the change of the populationsx(t) andy(t) canbe modeled by the equations

dxdt

= 1.6x−0.8y,

dydt

= 2.5x−1.2y.

? 1. What kind of interaction do we observe?


/ur de 13 13.pg

David opens a bank account with an initial balance of 500 dol-lars. Let b(t) be the balance in the account at timet. Thusb(0) = 500. The bank is paying interest at a continuous rate of4% per year. David makes deposits into the account at a contin-uous rate ofs(t) dollars per year. Suppose thats(0) = 500 andthats(t) is increasing at a continuous rate of 6% per year (Davidcan save more as his income goes up over time).(a) Set up a linear system of the form

dbdt

= m11b+m12s,

dsdt

= m21b+m22s.

m11 = ,m12 = ,m21 = ,m22 = .(b) Findb(t) ands(t).b(t) = ,s(t) = .


/ur de 13 14.pg


=[

-2 -22 -2

]x

with x(0) =[

11

].

Give your solution in real form.x1 = ,x2 = .

? 1. Describe the trajectory.


/ur de 13 15.pg


=[

-3 -36 3

]x

with x(0) =[

23

].

Give your solution in real form.x1 = ,x2 = .

? 1. Describe the trajectory.


/ur de 13 16.pg

Multiplying the differential equation

d fdt

+a f(t) = g(t),

wherea is a constant andg(t) is a smooth function, byeat, gives3

eat d fdt

+eata f(t) = eatg(t),

ddt

(eat f (t)

)= eatg(t),

eat f (t) =Z

eatg(t)dt,

f (t) = e−atZ

eatg(t)dt.

Use this to solve the initial value problemdxdt

=[

3 10 2

]x,

with x(0) =[

-54

],

i.e. find firstx2(t) and thenx1(t).x1(t) = ,x2(t) = .


4

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ2DirectionFields due 01/02/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 1-

/ur de 2 1.pg

Match the following equations with their direction field. Click-ing on each picture will give you an enlarged view. While youcan probably solve this problem by guessing, it is useful to tryto predict characteristics of the direction field and then matchthem to the picture. Here are some handy characteristics to startwith – you will develop more as you practice.

A. Sety equal to zero and look at how the derivative be-haves along thex-axis.

B. Do the same for they-axis by settingx equal to 0C. Consider the curve in the plane defined by settingy′ = 0

– this should correspond to the points in the picturewhere the slope is zero.

D. Settingy′ equal to a constant other than zero gives thecurve of points where the slope is that constant. Theseare called isoclines, and can be used to construct thedirection field picture by hand.

Go to this page to launch the phase plane plotter to check youranswers. (Choose the ”integral curves utility” from the appletmenu, enterdx/dt = 1 to identify the variablesx andt and thenenter the function you want fordy/dx= dy/dt = . . . ).

1. y′ =−2+x−y2. y′ = e−x−y3. y′ = 3sin(x)+1+y

A B C

2. (1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 2-

/ur de 2 2.pg

This problem is harder, and doesn’t give you clues as to whichmatches you have right. Study the previous problem, if you arehaving trouble.Go to this page to launch the phase plane plotter to check youranswers.Match the following equations with their direction field. Click-ing on each picture will give you an enlarged view.

1. y′ = 2x−1−y2

2. y′ = y(4−y)3. y′ = x+2y

4. y′ =y3

6−y− x3

6

A B

C D

3. (1 pt) rochesterLibrary/setDiffEQ2DirectionFields/ur de 2 3.pg

Match the following equations with their direction field. Click-ing on each picture will give you an enlarged view. While youcan probably solve this problem by guessing, it is useful to tryto predict characteristics of the direction field and then matchthem to the picture.Here are some handy characteristics to start with – you will de-velop more as you practice.





1. y′ = 2xy+2xe−x2

2. y′ = 2y−2

3. y′ =− (2x+y)(2y)

4. y′ = 2sin(3x)+1+y

1

A B

C D


Match the following equations with their direction field. Click-ing on each picture will give you an enlarged view. While youcan probably solve this problem by guessing, it is useful to tryto predict characteristics of the direction field and then matchthem to the picture. Here are some handy characteristics to startwith – you will develop more as you practice.





1. y′ = 2y+x2e2x

2. y′ =− (2x+y)(2y)

3. y′ = y+24. y′ = 2sin(x)+1+y

A B

C D


Use Euler’s method with step size 0.5 to compute the approxi-matey-valuesy1, y2, y3, andy4 of the solution of the initial-valueproblem

y′ =−2−2x−4y, y(0) = 2.

y1 = ,y2 = ,y3 = ,y4 = .


Use Euler’s method with step size 0.3 to estimatey(1.5), wherey(x) is the solution of the initial-value problem

y′ =−5x+y2, y(0) = 0.

y(1.5) = .


Suppose you have just poured a cup of freshly brewed coffeewith temperature 90C in a room where the temperature is 20C.Newton’s Law of Cooling states that the rate of cooling of anobject is proportional to the temperature difference between theobject and its surroundings. Therefore, the temperature of thecoffee,T(t), satisfies the differential equation

dTdt

= k(T−Troom)

whereTroom = 20 is the room temperature, andk is some con-stant.Suppose it is known that the coffee cools at a rate of 1C perminute when its temperature is 60C.A. What is the limiting value of the temperature of the coffee?limt→∞

T(t) =B. What is the limiting value of the rate of cooling?

limt→∞

dTdt

=C. Find the constantk in the differential equation.k = .D. Use Euler’s method with step sizeh = 3 minutes to estimatethe temperature of the coffee after 15 minutes.T(10) = .

2


3

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ3Separable due 01/03/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ3Separable/ur de 3 11.pg

Solve the separable differential equation

dydx

=−6y,

and find the particular solution satisfying the initial condition

y(0) = 4.

y(x) = .



dxdt

=4x,


x(0) = 1.

x(t) = .



dydt

=−8y6,


y(0) =−5.

y(t) = .



y′(x) =√

2y(x)+37,


y(−3) = 6.

y(x) = .



dydx

=−0.8cos(y)

,


y(0) =π4.

y(x) = .



dxdt

= x2 +19,


x(0) =−9.

x(t) = .

7. (1 pt) rochesterLibrary/setDiffEQ3Separable/osude 3 11.pg

Find the particular solution of the differential equation

dydx

= (x−8)e−2y

satisfying the initial conditiony(8) = ln(8).Answer:y = .Your answer should be a function ofx.



x2

y2−7dydx

=12y

satisfying the initial conditiony(1) =√

8.Answer:y = .Your answer should be a function ofx.

9. (1 pt) rochesterLibrary/setDiffEQ3Separable/jas74 5.pg

Findu from the differential equation and initial condition.

dudt

= e3.5t−1.6u, u(0) = 1.

u = .

10. (1 pt) rochesterLibrary/setDiffEQ3Separable/jas74 5a.pg

Solve the separable differential equation foru

dudt

= e4u+7t .

Use the following initial condition:u(0) = 9.u = .

11. (1 pt) rochesterLibrary/setDiffEQ3Separable/jas74 5b.pg

Solve the separable differential equation foru

dudt

= e3u+9t .

Use the following initial condition:u(0) =−8.u = .

12. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns74 10.pg


11x−4y√

x2 +1dydx

= 0.

Subject to the initial condition:y(0) = 7.y = .

1


Find f (x) if y = f (x) satisfies

dydx

= 30yx2

and they-intercept of the curvey = f (x) is 6.f (x) = .

14. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns74 13a.pg

Find an equation of the curve that satisfies

dydx

= 54yx5

and whosey-intercept is 5.y(x) = .


Find the solution of the differential equation

3e7x dydx

=−49xy2

which satisfies the initial conditiony(0) = 1.y = .


Find a functiony of x such that

6yy′ = x and y(6) = 11.

y = .


Find k such thatx(t) = 5t is a solution of the differential equa-

tiondxdt

= kx.

k = .


Solve the seperable differential equation

10yy′ = x.

Use the following initial condition:y(10) = 3.Expressx2 in terms ofy.x2 = (function of y).


Solve the differential equation

(y12x)dydx

= 1+x.

Use the initial conditiony(1) = 5.Expressy13 in terms ofx.y13 = .


Solve the seperable differential equation for.

dydx

=1+xxy7 ; x > 0

Use the following initial condition:y(1) = 5.y8 = .

21. (1 pt) rochesterLibrary/setDiffEQ3Separable/ns74 8b.pg

Find the functiony = y(x) (for x > 0 ) which satisfies the sepa-rable differential equation

dydx

=7+20x

xy2 ; x > 0

with the initial conditiony(1) = 6.y = .


Find the solution of the differential equation

(ln(y))7 dydx

= x7y

which satisfies the initial conditiony(1) = e2.y = .


A. Solve the following initial value problem:

(t2−18t +72)dydt

= y

with y(9) = 1. (Findy as a function oft.)y = .B. On what interval is the solution valid?Answer: It is valid for < t < .C. Find the limit of the solution ast approaches the left end ofthe interval.(Your answer should be a number or the word ”infinite”.)Answer: .D. Similar to C, but for the right end.Answer: .


The differential equation

dydx

= cos(x)y2 +10y+25

5y+25

has an implicit general solution of the formF(x,y) = K.In fact, because the differential equation is separable, we candefine the solution curve implicitly by a function in the form

F(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) =

.

2



36dydx

= (16−x2)−1/2 exp(−6y)


F(x,y) = G(x)+H(y) = K.


.



dydx

=14

y1/8 +4x2y1/8


F(x,y) = G(x)+H(y) = K.


.



(15+8cos(x))dydx

= sin(x) cos(y)


F(x,y) = G(x)+H(y) = K.


.


A. Find y in terms ofx if

dydx

= x2y−3

andy(0) = 8.y(x) = .B. For whatx-interval is the solution defined?(Your answers should be numbers or plus or minus infinity. Forplus infinity enter ”PINF”; for minus infinity enter ”MINF”.)The solution is defined on the interval:

< x < .



dydx

=6x+6

15y2 +6y+6


F(x,y) = G(x)+H(y) = K.


.



exp(y)dydx

=14x+2

2 sin(y)+8 cos(y)


F(x,y) = G(x)+H(y) = K.


.


A. Solve the following initial value problem:

cos(t)2 dydt

= 1

with y(21) = tan(21).(Findy as a function oft.)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 ast approaches the left end ofthe interval. (Your answer should be a number or ”PINF” or”MINF”.”PINF” stands for plus infinity and ”MINF” stands for minusinfinity.)Answer: .D. Similar to C, but for the right end.Answer: .

3



dydx

= 8+6x+16y+12xy

has an implicit general solution of the formF(x,y) = K.

In fact, because the differential equation is separable, we candefine the solution curve implicitly by a function in the form

F(x,y) = G(x)+H(y) = K.


.


4

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ4Linear1stOrder due 01/04/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/osu de 4 14.pg


dydx

+3y = 7

satisfying the initial conditiony(0) = 0.Answer:y = .Your answer should be a function ofx.

2. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 16.pg

Find the function satisfying the differential equation

f ′(t)− f (t) =−2t

and the conditionf (3) = 3.f (t) = .


GUESS one functiony(t) which solves the problem below, bydetermining the general form the function might take and thenevaluating some coefficients.

7tdydt

+y = t3

Findy(t).y(t) = .


GUESS one functiony(t) which solves the problem below, bydetermining the general form the function might take and thenevaluating some coefficients.

dydt

+7y = exp(2t)

Findy(t).y(t) = .


Find the function satisfying the differential equation

y′−5y = 6e8t

andy(0) =−1.y = .


Solve the following initial value problem:

tdydt

+9y = 4t

with y(1) = 1.y = .


Solve the following initial value problem:dydt

+0.7ty = 8t

with y(0) = 8.y = .


Solve the initial value problem

10(t +1)dydt−9y = 9t,

for t >−1 with y(0) = 3.y = .



dxdt−3x = cos(5t)

with x(0) = 5.x(t) = .

10. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/osu de 4 15.pg


dydx

+ycos(x) = 5cos(x)

satisfying the initial conditiony(0) = 7.Answer:y(x)= .


Solve the initial value problemdydt−y = 3exp(t)+21exp(4t)

with y(0) = 7.y = .


Solve the initial value problemdydt

+2y = 40sin(t)+25cos(t)

with y(0) = 7.y = .


Solve the following initial value problem:

9dydt

+y = 63t

with y(0) = 7.(Findy as a function oft.)y = .



8(sin(t)dydt

+(cost)y) = (cos(t))(sin(t))6,

for 0 < t < π andy(π/2) = 16.y = .

1


Find the functiony(t) that satisfies the differential equation

dydt−2ty =−6t2et2

and the conditiony(0) = 5.y(t) = .


A. Let g(t) be the solution of the initial value problem

5tdydt

+y = 0, t > 0,

with g(1) = 1.Findg(t).g(t) = .B. Let f (t) be the solution of the initial value problem

5tdydt

+y = t5

with f (0) = 0.Find f (t).f (t) = .C. Find a constantc so that

k(t) = f (t)+cg(t)

solves the differential equation in part B andk(1) = 14.c = .


A. Let g(t) be the solution of the initial value problemdydt

+7y = 0,

with y(0) = 1.Findg(t).g(t) = .B. Let f (t) be the solution of the initial value problem

dydt

+7y = exp(1t)

with y(0) = 1/8.Find f (t).f (t) = .C. Find a constantc so that

k(t) = f (t)+cg(t)

solves the differential equation in part B andk(0) = 12.c = .


Find a family of solutions to the differential equation

(x2−4xy)dx+xdy= 0

(To enter the answer in the form below you may have to rear-range the equation so that the constant is by itself on one side ofthe equation.) Then the solution in implicit form is:the set of points (x, y) whereF(x,y) =

= constant


A functiony(t) satisfies the differential equation

dydt

=−y4−2y3 +15y2.

(a) What are the constant solutions of this equation?Separate your answers by commas.

.(b) For what values ofy is y increasing?

< y < .


A Bernoulli differential equation is one of the form

dydx

+P(x)y = Q(x)yn.

Observe that, ifn = 0 or 1, the Bernoulli equation is linear.For other values ofn, the substitutionu = y1−n transforms theBernoulli equation into the linear equation

dudx

+(1−n)P(x)u = (1−n)Q(x).

Use an appropriate substitution to solve the equation

xy′+y = 2xy2,

and find the solution that satisfiesy(1) = 5.y(x) = .

21. (1 pt) rochesterLibrary/setDiffEQ4Linear1stOrder/ur de 4 18a.pg


dydx

+P(x)y = Q(x)yn (∗)

Observe that, ifn = 0 or 1, the Bernoulli equation is linear.For other values ofn, the substitutionu = y1−n transforms theBernoulli equation into the linear equation

dudx

+(1−n)P(x)u = (1−n)Q(x).

Consider the initial value problem

xy′+y =−3xy2, y(1) = 4.

(a) This differential equation can be written in the form(∗) withP(x) = ,Q(x) = , andn = .(b) The substitutionu = will transform it into the linearequationdudx

+ u = .

(c) Using the substitution in part (b), we rewrite the initial con-dition in terms ofx andu:u(1) = .(d) Now solve the linear equation in part (b). and find the solu-tion that satisfies the initial condition in part (c).u(x) = .(e) Finally, solve fory.y(x) = .

2



dydx

+P(x)y = Q(x)yn.

Observe that, ifn = 0 or 1, the Bernoulli equation is linear.For other values ofn, the substitutionu = y1−n transforms the

Bernoulli equation into the linear equation

dudx

+(1−n)P(x)u = (1−n)Q(x).

Use an appropriate substitution to solve the equation

y′− 9x

y =y5

x8 ,

and find the solution that satisfiesy(1) = 1.y(x) = .


3

Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ5ModelingWith1stOrder due 01/05/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ5ModelingWith1stOrder-

/ur de 5 9.pg

A curve passes through the point(0,3) and has the property thatthe slope of the curve at every pointP is twice they-coordinateof P. What is the equation of the curve?y(x) =


/ns7 4 31.pg

A tank contains 1240 L of pure water. A solution that contains0.03 kg of sugar per liter enters a tank at the rate 9 L/min Thesolution is mixed and drains from the tank at the same 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 tankafter 90 minutes.concentration =


/ns7 4 31a.pg

A tank contains 2040 L of pure water. A solution that contains0.06 kg of sugar per liter enters tank at the rate 7 L/min. Thesolution 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 Srepresenting the amount of sugar (in kg) at time t (inminutes) write a differential equation which models this situa-tion.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 84 minutes.S(84) = (include units)

Click here for help with units


/ns7 4 31b.pg

A tank contains 1000 L of pure water. Solution that contains0.03 kg of sugar per liter enters the tank at the rate 8 L/min, andis thoroughly mixed into it. The new solution drains out of thetank 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)

(c) As t becomes large, what value isy(t) approaching ? In otherwords, calculate the following limit. lim

t→∞y(t) = (kg)


/ns7 4 31c.pg

A tank contains 100 kg of salt and 2000 L of water. A solutionof a concentration 0.025 kg of salt per liter enters a tank at therate 7 L/min. The solution is mixed and drains from the tank atthe same rate.(a) What is the concentration of our solution in the tank initially?concentration = (kg/L)(b) Find the amount of salt in the tank after 4.5 hours.amount = (kg)(c) Find the concentration of salt in the solution in the tank astime approaches infinity.concentration = (kg/L)


/ns7 4 31d.pg

A tank contains 100 kg of salt and 1000 L of water. Pure waterenters a tank at the rate 12 L/min. The solution is mixed anddrains from the tank at the rate 6 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 2 hours.amount = (kg)(c) Find the concentration of salt in the solution in the tank astime approaches infinity. (Assume your tank is large enough tohold all the solution.)concentration = (kg/L)


/ns7 4 31e.pg

A tank contains 1820 L of pure water. A solution that contains0.07 kg of sugar per liter enters tank at the rate 9 L/min Thesolution 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) aftert minutes.y(t) = (function oft)(b) Find the amout of the sugar after 42 minutes.y(42) = (include units)


/ns7 5 2.pg

A cell of some bacteria divides into two cells every 20 min-utes.The initial population is 2 bacteria.(a) Find the size of the population after t hoursy(t) =(function oft)(b) Find the size of the population after 4 hours.y(4) =(c) When will the population reach 8?

1

T =


/ur de 5 7.pg

A cell of some bacteria divides into two cells every 40 minutes.The initial population is 200 bacteria.(a) Find the population aftert hoursy(t) = (function oft)(b) Find the population after 2 hours.y(2) =(c) When will the population reach 400?T =


/ns7 5 3.pg

A bacteria culture starts with 160 bacteria and grows at a rateproportional to its size. After 4 hours there will be 640 bacteria.(a) Express the population aftert hours as a function oft.population: (function of t)(b) What will be the population after 7 hours?

(c) How long will it take for the population to reach 2930 ?


/ns7 6 1.pg

A populationP obeys the logistic model. It satisfies the equationdPdt

=9

1100P(11−P) for P > 0.

(a) The population is increasing when < P <(b) The population is decreasing whenP >(c) Assume thatP(0) = 4. FindP(84).P(84) =


/ur de 5 10.pg

Suppose that a population develops according to the logisticequation

dPdt

= 0.25P−0.0025P2

wheret is measured in weeks.(a) What is the carriying capacity?(b) Is the solution increasing or decreasing whenP is between 0and the carriying capacity??(c) Is the solution increasing or decreasing whenP is greaterthan the carriying capacity??


/ur de 5 11.pg

Biologists stocked a lake with 300 fish and estimated the carry-ing capacity (the maximal population for the fish of that speciesin that lake) to be 9800. The number of fish doubled in the firstyear.(a) Assuming that the size of the fish population satisfies thelogistic equation

dPdt

= kP

(1− P

K

),

determine the constantk, and then solve the equation to find anexpression for the size of the population aftert years.k = ,P(t) = .(b) How long will it take for the population to increase to 4900(half of the carrying capacity)?It will take years.


/ur de 5 12.pg

Another model for a growth function for a limited pupulationis given by the Gompertz function, which is a solution of thedifferential equation

dPdt

= cln

(KP

)P

wherec is a constant andK is the carrying capacity.(a) Solve this differential equation forc = 0.2, K = 5000, andinitial populationP0 = 400.P(t) = .(b) Compute the limiting value of the size of the population.limt→∞

P(t) = .

(c) At what value ofP doesP grow fastest?P = .


/ns7 5 10.pg

An unknown radioactive element decays into non-radioactivesubstances. In 140 days the radioactivity of a sample decreasesby 77 percent.(a) What is the half-life of the element?half-life: (days)(b) How long will it take for a sample of 100 mg to decay to 41mg?time needed: (days)


/ur de 5 2.pg

A body of mass 6 kg is projected vertically upward with an ini-tial velocity 61 meters per second.The gravitational constant isg = 9.8m/s2. The air resistance isequal tok|v| wherek is a constant.Find a formula for the velocity at any time ( in terms ofk ):v(t) =Find the limit of this velocity for a fixed timet0 as the air resis-tance coefficientk goes to 0. (Entert0 as t0 .)v(t0) =How does this compare with the solution to the equation for ve-locity when there is no air resistance?This illustrates an important fact, related to the fundamental the-orem of ODE and called ’continuous dependence’ on parame-ters 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 timet.The fact that the terminal timet under consideration is a fixed,finite number is important. If you consider ’infinite’t, or the

2

’final’ result you may get very different answers. Consider forexample a solution toy′ = y, whose initial condition is essen-tially zero, but which might vary a bit positive or negative. Ifthe initial condition is positive the ”final” result is plus infin-ity, but if the initial condition is negative the final condition isnegative infinity.


/ur de 5 8.pg

You have 600 dollars in your bank account. Suppose yourmoney is compounded every month at a rate of 0.5 percent permonth.(a) How much do you have aftert years?y(t) = (function oft)(b) How much do you have after 110 months?y(110) =


/ur de 5 1.pg

A young person with no initial capital investsk dollars per yearin a retirement account at an annual rate of return 0.1. Assumethat investments are made continuously and that the return iscompounded continuously.Determine a formula for the sumS(t) – (this will involve theparameterk):S(t) =What value ofk will provide 1589000 dollars in 47 years?k =


/ur de 5 3.pg

Here is a somewhat realistic example which combines the workon earlier problems. You should use the phase plane plotter tolook at some solutions graphically before you start solving thisproblem and to compare with your analytic answers to help youfind errors. You will probably be surprised to find how long ittakes to get all of the details of solution of a realistic problemright, even when you know how to do each of the steps.There are 1300 dollars in the bank account at the beginning ofJanuary 1990, and money is added and withdrawn from the ac-count at a rate which follows a sinusoidal pattern, peaking inJanuary and in July with money being added at a rate corre-sponding to 2110 dollars per year, while maximum withdrawalstake place at the rate of 490 dollars per year in April and Octo-ber.The interest rate remains constant at the rate of 6 percent peryear, compounded continuously.Let y(t) represents the amount of money at timet (t is in years).y(0) = (dollars)Write a formula for the rate of deposits and withdrawals (usingthe functions sin(), cos() and constants):g(t) =The interest rate remains constant at 6 percent per year over thisperiod of time.With y representing the amount of money in dollars at timet (inyears) write a differential equation which models this situation.

y′ = f (t,y) =

.Note: Usey rather thany(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 at timet wheret is the number of years since January 1990.y(t) =Find the amount of money in the bank at the beginning of Janu-ary 2000 (10 years later):Find a solution to the equation which does not become infinite(either positive or negative) over time:y(t) =During which months of the year does this non-growing solutionhave the highest values??


/ur de 5 13.pg

How long will it take an investment to double in value if theinterest rate is 4% compounded continuously?Answer: years.


/ur de 5 4.pg

Newton’s law of cooling states that the temperature of an ob-ject changes at a rate proportional to the difference between itstemperature and that of its surroundings. Suppose that the tem-perature of a cup of coffee obeys Newton’s law of cooling. Ifthe coffee has a temperature of 210 degrees Fahrenheit whenfreshly poured, and 1 minutes later has cooled to 199 degreesin a room at 74 degrees, determine when the coffee reaches atemperature of 159 degrees.The coffee will reach a temperature of 159 degrees in min-utes.


/ur de 5 14.pg

A thermometer is taken from a room where the temperature is25oC to the outdoors, where the temperature is 4oC. After oneminute the thermometer reads 19oC.(a) What will the reading on the thermometer be after 5 moreminutes?

,(b) When will the thermometer read 5oC?

minutes after it was taken to the outdoors.


/ur de 5 5.pg

Susan finds an alien artifact in the desert, where there are tem-perature variations from a low in the 30s at night to a high inthe 100s in the day. She is interested in how the artifact willrespond to faster variations in temperature, so she kidnaps theartifact, takes it back to her lab (hotly pursued by the militarypolice who patrol Area 51), and sticks it in an ”oven” – that is,a closed box whose temperature she can control precisely.

3

Let T(t) be the temperature of the artifact. Newton’s law ofcooling says thatT(t) changes at a rate proportional to the dif-ference between the temperature of the environment and thetemperature of the artifact. This says that there is a constantk, not dependent on time, such thatT ′ = k(E−T), whereE isthe temperature of the environment (the oven).Before collecting the artifact from the desert, Susan measuredits temperature at a couple of times, and she has determined thatfor the alien artifact,k = 0.7.Susan preheats her oven to 90 degrees Fahrenheit (she has stub-bornly refused to join the metric world). At timet = 0 the ovenis at exactly 90 degrees and is heating up, and the oven runsthrough a temperature cycle every 2π minutes, in which its tem-perature varies by 25 degrees above and 25 degrees below 90degrees.Let E(t) be the temperature of the oven aftert minutes.E(t) =At time t = 0, when the artifact is at a temperature of 35 de-grees, she puts it in the oven. LetT(t) be the temperature of theartifact at timet. ThenT(0) = (degrees)Write a differential equation which models the temperature ofthe artifact.T ′ = f (t,T) =

.Note: UseT rather thanT(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 it help-ful to know that ifa is a constant, thenZ

sin(t)eatdt =1

a2 +1eat(asin(t)−cos(t))+C.

T(t) =After Susan puts in the artifact in the oven, the military policebreak in and take her away. Think about what happens to herartifact ast → ∞ and fill in the following sentence:For large values oft, even though the oven temperature variesbetween 65 and 115 degrees, the artifact varies from to

degrees.


/ur de 5 6.pg

Here is a multipart example on finance. Be patient and carefulas you work on this problem. You will probably be surprised tofind how long it takes to get all of the details of the solution of arealistic problem exactly right, even when you know how to doeach of the steps. Use the computer to check the steps for youas you go along. There is partial credit on this problem.A recent college graduate borrows 100000 dollars at an (an-nual) interest rate of 9 per cent. Anticipating steady salary in-creases, the buyer expects to make payments at a monthly rateof 600(1+ t/160) dollars per month, wheret is the number ofmonths since the loan was made.Let y(t) be the amount of money that the graduate owest monthsafter the loan is made.

y(0) = (dollars)With y representing the amount of money in dollars at timet (inmonths) write a differential equation which models this situa-tion.y′ = f (t,y) =

.Note: Usey rather thany(t) since the latter confuses the com-puter. Remember to check your units, but don’t enter units forthis equation – the computer won’t understand them.

0Find an equation for the amount of money owed aftert months.y(t) =Next we are going to think about how many months it will takeuntil the loan is paid off. Remember thaty(t) is the amount thatis owed aftert months. The loan is paid off wheny(t) =Once you have calculated how many months it will take topay 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 orusea 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 fory (usingxas 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 20years, with the same payment plan as above, how much couldbe borrowed?Borrowed amount =


/ur de 5 15.pg

In the circuit shown in the figure above a battery supplies a con-stant voltage ofE = 40V, the inductance isL = 2H, the resis-tance isR= 30Ω, andI(0) = 0. Find the current aftert seconds.I(t) = .

4


/ur de 5 16.pg

In the circuit shown in the figure above a generator supplies avoltage ofE(t) = 60sin(50t)V, the inductance isL = 2H, theresistance isR = 10Ω, and I(0) = 1. Find the current aftertseconds.I(t) = .


/ur de 5 17.pg

The figure above shows a circuit containing an electromotiveforce, a capacitor with a capacitance ofC farads(F), and a re-sistor with a resistance ofR ohmsΩ. The voltage drop across

the capacitor isQ/C, whereQ is the charge (in coulombs), so inthis case Kirchhoff’s Law gives

RI+QC

= E(t).

SinceI =dQdt

, we have

RdQdt

+1C

Q = E(t).

Suppose the resistance is 20Ω, the capacitance is 0.2F, a bat-tery gives a constant voltage of 50V, and the initial charge isQ(0) = 0C.Find the charge and the current at timet.Q(t) = ,I(t) = .


/ur de 5 18.pg

Let P(t) be the performance level of someone learning a skill

as a function of the training timet. The derivativedPdt

repre-

sents the rate at which performance improves. IfM is the maxi-mum level of performance of which the learner is capable, thena model for learning is given by the differential equation

dPdt

= k(M−P(t))

wherek is a positive constant.Two new workers, Andy and Bob, were hired for an assemblyline. Andy could process 12 units per minute after one hourand 13 units per minute after two hours. Bob could process 11units per minute after one hour and 14 units per minute after twohours. Using the above model and assuming thatP(0) = 0, esti-mate the maximum number of units per minute that each workeris capable of processing.Andy: ,Bob: .


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Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ6AutonomousStability due 01/06/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ6AutonomousStability-

/ur de 6 1.pg

The graph of the functionf (x) is

(the hori-zontal axis is x.)Consider the differential equationx′(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.

?

?

?

?


/ur de 6 2.pg

The graph of the functionf (x) is

(the hori-zontal axis is x.)

Given the differential equationx′(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.

?

?

?

?


/ur de 6 3.pg

Given the differential equationx′ = −(x+ 3.5) ∗ (x+ 2)3(x+0.5)2(x−1).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. )

?

?

?

?


/ur de 6 4.pg

Given the differential equationx′(t) = x4+2x3−5.5x2−0.5x+1.3125.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. )

?

?

?

?

1


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Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ7Exact due 01/07/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ7Exact/ur de 7 1.pg

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

ydy−xdx= 0

F(x,y) =


Use the ”mixed partials” check to see if the following differen-tial equation is exact.If it is exact find a function F(x,y) whose level curves are solu-tions to the differential equation

(4x4 +2y)dx+(4x+3y4)dy= 0

?F(x,y) =



(−1xy2−3y)dx+(−1x2y−3x)dy= 0

?F(x,y) =



dydx

=+4x2 +2y−2x+4y1

?F(x,y) =



(3ex sin(y)−4y)dx+(−4x+3ex cos(y))dy= 0

?F(x,y) =


Check that the equation below is not exact but becomes exactwhen 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) =


Find an explicit or implicit solutions to the differential equation

(x2 +2xy)dx+xdy= 0

F(x,y) =


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Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ8FundTheorem due 01/08/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ8FundTheorem/ur de 8 1.pg

This problem involves using the uniqueness property (from the Fundamental Theorem of ordinary differential equations.) It can’t begraded 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 ify1(t) is a solution to the differential equationy′(t) = y(t), theny2(t) =y1(t +a) is also a solution to the differential equation. (You will need to use the known facts abouty1 to calculate thaty′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 ofy1 andy2 and using a sketch of the direction field explain why it is obviousthat if y1 is a solution theny2 has to be a solution also.

D. Describe in words why ify1(t) is any solution to the differential equationy′ = f (y) theny2(t) = y1(t +a) is also a solution.

E. Show that ify1(t) solvesy′(t) = y(t), theny2(t) = Ay1(t) also solves the same equation.

F. Suppose thaty1(t) solvesy′(t) = y(t) andy(0) = 1. (Such a solution is guaranteed by the fundamental theorem.). Lety2(t) =y1(t +a) and lety3(t) = y1(a)y1(t). Calculate the valuesy2(0) andy3(0). Use the uniqueness property to show thaty2(t) = y3(t) forall t.

G. Explain how this proves that any solution toy′ = 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 ifz= 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) = 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 plotterhere or at thedirection field plotter page. Choose ”integral curves utility” from the ”mainscreen” menu of xFunctions to get to the phaseplane plotter.



A. Using the same technique as in the previous problem show that if a functiony1(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 plotterhere or at thedirection field plotter page. Choose ”integral curves utility” from the ”mainscreen” menu of xFunctions to get to the phaseplane plotter.

1



Use the same ideas as in the previous problems.A. Suppose thaty1(t) satisfies the equationy′′ + y = 0 andy1(0) = 0andy′1(0) = 1. Such a function exists because of the

fundamental theorem. (We all know that it issin(t), but you should not use that fact in answering the questions below.)Show thaty2(t) = y′1(t) also satisfies the equationy′′+y = 0 and thaty2(0) = 1 andy′2(0) = 0.

B. If y3(t) = y′2(t) show, using the uniqueness property, thaty3(t) =−y1(t)

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

D. Use the uniqueness property to show thaty1(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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Arnie Pizer Rochester Problem Library Fall 2005WeBWorK assignment DiffEQ9Linear2ndOrderHomog due 01/09/2007 at 02:00am EST.

1. (1 pt) rochesterLibrary/setDiffEQ9Linear2ndOrderHomog-

/ur de 9 1.pg

Findy as a function oft if

4y′′−729y = 0,

y(0) = 7, y′(0) = 2.y(t) =


/ur de 9 6.pg


10000y′′−9y = 0

with y(0) = 3, y′(0) = 6.y =


/ur de 9 8.pg


10000y′′+729y = 0,

y(0) = 7, y′(0) = 4.y =


/ur de 9 2.pg


y′′−9y′ = 0,

y(0) = 3, y(1) = 3.y(t) =Remark: The initial conditions involve values at two points.


/ur de 9 7.pg


36y′′−12y′+y = 0,

y(0) = 7, y′(0) = 3.y =


/ur de 9 3.pg


6y′′+32y = 0,

y(0) = 7, y′(0) = 2.y(t) =Note: This particular weBWorK problem can’t handle complexnumbers, so write your answer in terms of sines and cosines,rather than using e to a complex power.


/ur de 9 4.pg

Findy as a function oft if36y′′+72y′+56y = 0,

y(0) = 7, y′(0) = 2.y =Note: This problem cannot interpret complex numbers. Youmay need to simplify your answer before submitting it.


/ur de 9 5.pg


y(0) = 7, y′(0) = 2.y(t) =Note: This problem cannot interpret complex numbers. Youmay need to simplify your answer before submitting it.


/ur de 9 11.pg

Findy as a function oft ify′′+6y′+25y = 0, y(0) = 4, y′(0) = 4.

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


/ur de 9 9.pg

Findy as a function oft if81y′′+18y′ = 0,

y(0) = 1, y′(0) = 2.y =Note: This problem cannot interpret complex numbers. Youmay need to simplify your answer before submitting it.


/ur de 9 10.pg

Find the functiony1 of t which is the solution of

100y′′+160y′+15y = 0

with initial conditions y1(0) = 1, y′1(0) = 0.y1 =Find the functiony2 of t which is the solution of

100y′′+160y′+15y = 0

with initial conditions y2(0) = 0, y′2(0) = 1.y2 =Find the Wronskian

W(t) = W(y1,y2).

W(t) =1

Remark: You can find W by direct computation and use Abel’stheorem as a check. You should find that W is not zero and soy1 andy2 form a fundamental set of solutions of

100y′′+160y′+15y = 0.


/ur de 9 12.pg

Findy as a function oft if1296y′′−360y′+25y = 0,

y(0) = 2, y′(0) = 4.y =


/ur de 9 13.pg


y(3) = 4, y′(3) = 9.y =


/ur de 9 14.pg

Determine whether the following pairs of functions are linearlyindependent or not.

? 1. f (t) = t2 +9t andg(t) = t2−9t

? 2. The Wronskian of two functions isW(t) = t are thefunctions linearly independent or dependent?

? 3. f (t) = t andg(t) = |t|


/ur de 9 15.pg

Suppose that the Wronskian of two functionsf1(t) and f2(t) is

given byW(t) = t2−4= det

[f1(t) f2(t)f ′1(t) f ′2(t)

]Even though you

don’t know the functionsf1 and f2 you can determine whetherthe 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) = 0a f ′1(2)+b f ′2(2) = 0

have more than one solution.? 3. The equations

a f1(0)+b f2(0) = ca f ′1(0)+b f ′2(0) = d

have a unique solution for anyc andd

? 4. The equations

a f1(2)+b f2(2) = ca f ′1(2)+b f ′2(2) = d

have a unique solution for anyc andd? 5. The vectors( f1(0), f ′1(0)) and ( f2(0), f ′2(0)) are lin-

early independent


/ur de 9 16.pg

Determine which of the following pairs of functions are linearlyindependent.

? 1. f (t) = eλt cos(µt) , g(t) = eλt sin(µt) ,µ 6= 0

? 2. f (θ) = cos(3θ) , g(θ) = 2cos3(θ)−4cos(θ)? 3. f (t) = 2t2 +14t , g(t) = 2t2−14t

? 4. f (x) = x2 , g(x) = 4|x|2


/ur de 9 17.pg

Match the second order linear equations with the Wronskian of(one of) their fundamental solution sets.

1. y′′− ln(t)y′−9y = 02. y′′+4y′−9y = 03. y′′− 1

t y′−9y = 0, t > 04. y′′− 2

t y′−9y = 05. y′′+ 2

t y′−9y = 0

A. W(t) = 2e−4t

B. W(t) = t2

C. W(t) = et ln(t)−t

D. W(t) = 5t2

E. W(t) = 7t


/ur de 9 18.pg


x2y′′+2xy′−12y = 0,

y(1) =−4, y′(1) =−10.y =


/ur de 9 19.pg


x2y′′−7xy′+16y = 0,

y(1) = 5, y′(1) = 3.y =


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Arnie Pizer Rochester Problem Library Fall...

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