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Differential Equations and Linear Algebra 2nd edition Jerry Farlow, James E. Hall, Jean Marie McDill, Beverly H. West - Solutions

The simplest case of the logistic model is represented by the DE dy/dt = ky( I - y). Where k > 0 is the growth rate constant Draw a direction field for this equation when k = I. Find the constant solutions. Explain why this model represents limited growth. What happens in the long run (that is, as
When a lint-order DE has the form y' = f(y), so the right-band side doesn't depend oo t, the equation is called autonomous (which means independent of time).The logistic equation y' = ky(1 - y) in Problem 60 and the equation y' = y2 - 4 in Example 9 are examples of autonomous equations.(a) List
Explore the direction fields of the DEsy' = y2, y' = (y + l )2 , and y' = y2 + 1 .Describe their similarities and differences. Then answer the following questions:(a) Suppose each equation has initial condition y(0) = I Is one solution larger than the other for t > 0?(b) You can verify that y = 1
Determine the basin of attraction for each constant solution of the autonomous equations in Problems. That is, sketch a direction field and highlight the equilibrium solutions. For each, color the portion of the plane from which solutions are attracted to that equilibrium. a. y' = y( l - y) b. y' =
For the DEs in Problems, use appropriate software to draw direction fields. Then discuss what you can deduce about their solutions by sketching some representative solutions folk>wing the direction field. Include such features as constant and period solutions, special solutions to which other
Verify that y = cet· is a solution for any real c. of y' = 2ty determine c so that y(0) = 2
Verify that the function y = et cos t + cet is a solution for every real c. of the DE y'- y = -et sin t. Determine c so that y(0) = - l
For the IJEs in Problems use the corresponding direction field to draw some solution. Tryto give the general solution as formulas the substitute your guesses into the DE to see if you got them right. Where you did not explain ita. y' = 2yb. y' = -t/y
Use separation of variables to obtain solutions to the DEs and JVPs in Problems. Solve for y when possible. a. y' = t2/y b. ty' = √1-y2
Problems involve a variety of integration techniques. Recall that the formula for integration by parts is ∫u dv = uv - ∫b du, where the variables u and dv must be assigned carefully. Determine the solutions to the following DEs. a . y' = (cos2 y) lnt b. y' = (t2 - 5) cos2t
Solve the DEs in Problems. Most of these solutions will require the use of the method of partial fractions. (See Appendix PF for a review of this material.) Write down the equilibrium solutions as well as the non-equilibrium solutions for each DE.a. y' = 1 - y2b. y' = 2y - y2
For each DE in problems solve analytically to obtain solution curves through the point (1,1) and (-1,-1), Then using an appropriate software package draw the direction field and superimpose your solution curves onto it. a. dy/dt = t b. dy / dt = cos t
Making Equations Separable Many differential equations that are not separable can be made separable by making a proper substitution. One example is the class of first-order equations with right-hand sides that are functions of the combination y/r (or r/ y1. Given such a DECalled Euler-homogenous.
Given the differential equation y' = f (at + by + c), it can be shown that the substitution u = at + by + c, where a, b, and c are constants, will transform the differential equation into the separable equation u' = a + bf(u). Use this substitution to solve the DEs in Problems. a. y' = (y+t)2 b. y'
When a one-parameter family of curves satisfies a first-order DE, we can find another such family as solution curves of a related DE with the property that a curve of the other family orthogonally. Each family constitutes the set of orthogonal trajectories for the other.For the following questions
Use the problem 48 to determine the family of trajectories or thogonal to each given family in Problem.In problem 48When a one-parameter family of curves satisfies a first-order DE, we can find another such family as solution curves of a related DE with the property that a curve of the other family
With the help of suitable computer software, fore problems graph the families of curves along with their families of orthogonal trajectories. a. y = c (horizontal lines) b. 4x2 + y2 = c (ellipses)
The sine function sin x has the property that the square of itself plus the square of its derivative is identically equal to one. Find the most general function that has this property.
The rate at which the volume of a mothball evaporates from solid to gas is proportional to the surface area of the ball. Suppose a mothball has been observed at one time to have a radius of 0.5 in. and, six months later, a radius of 0.25 in. (a) Express the radius of the mothball as a function of
Four bugs sit at the comers of a square carpet L in. on a side. Simultaneously, each starts walking at the same rate of I in/sec toward the bug on its right See Fig. 1.3.9(a).(a) Show that the bugs collide at the center of the carpet in exactly L sec.(b) Using the result from (a). but using no
For the NP y' = t/y, y(0) = I, (a) Find Euler-approximate solution values at t = 0.1, t = 0.2 and t = 0.3 with h = 0.1 (b) Repeat (a) with h = 0.05. (c) Compute an analytic solution y(t), and compare the values of y(0.2) with your results from (a) and (b).
An interesting analysis results from playing with the equation of stefan's law. For dT/dt = k(M4-T4), let k = 0.05, M = 3 T(0) = 4 (a) Estimate T(1) by Euler's method with step sizes h = 0.25, h = 0.1 (b) Graph a direction field and both multistep approximations from (a). Explain why and how the
Nasty Surprise Use Euler's method with h = 0.25 to approximate the solution of y' = y2, y(0) = 1, at t = 0.25, t = 0.50, t = 0.75, and t = 1. Verify that the exact solution is y(t) = 1/(1-t); does this help explain what happened to the Euler approximation?
Obtain an estimate for the value of e by using Euler's method to approximate the solution of the IVP y' = y, y(0) = 1, at t = 1, using smaller and smaller values of h. As h decreases, the approximation for e gets better for a while but will eventually worsen, due to round off when h is small enough
The initial-value problem y' = y1/3, y(0) = 0, has an infinite number of solution two of which are y(t) = 0 and y(t) = (2t/3)3/2, These solutions are drawn in Fig. 1.4.7; the nonzero solution is tangent to the t-axis at the origin.(a) What happens if Euler's method is applied to this problem? (b)
The solution of the IVP y' = y, y(0) = A, is y(t) = Aet. If a round off error of ɛ occurs when the value of A is entered, how will this affect the solution at t = 1? What about at t=10 and t = 20?
We considered the DE y' = y2 - 4 in Example 8 of Sec. 1.2. (The direction field is graphed in Fig. 1.2.5.) What is the result of applying Euler's method (or other methods, if available) to the IVPsy' = y2 - 4, y(0) = 2 and y' = y2 - 4, y(0) = -2?How accurate are your approximations? What should the
The fourth-order approximation invented by Runge and Kutta can be surprisingly accurate, even with a ridiculously large step size. To see this, for Problem, use the given step size with the IVPy' = t+y , y(0) = 0(a) Compute for a single step the Euler approximation, the second-order Runge-Kutta
Use the Runge-Kuria method to approximate solutions to the following NPs and compare the results with the results obtained earlier by Euler's method (and with exact values when possible). You may wish to implement your solutions using a spreadsheet program, with several more columns for the
We will investigate the local discretization error in applying the Euler approximation given by equations (5) and (6).(a) If y(t) is the exact solution of y' = f(t. y) use the chain rule to calculate y "(t) and explain why it is continuous.(b) Recall the following from calculus: Remember that y(tn
(a) Replace the so-called two-term Taylor estimate (equation (8) in Problem 23) by the three-term result:Compute the second derivative y" in terms of f, and fy (partial derivatives of f, with respect to t and y, respectively), and deduce the three-term Taylor approximation: (b) Show that the local
In Problems, approximate the solution to the IVP at t = 0.2 by richardson's extrapolation using Euler's method with h = 0.1 and h/2 = 0.05. Compare with exact solutions when possible. a. y' = y, y(0) = 1 b. y' = ty, y(0) = 1
(a) Show that the lVP y' = j(t, y), y( to) = yo is equivalent to the integral equationby verifying the following two statements: (i) Every solution y( t) of the IVP satisfies the integral equation; (ii) Any function y(t) satisfying the integral equation satisfies the lVP. (b) Convert the IVP y' =
If you have access to software with other methods, choose one or two of Problems and make a study (for fixed step size) of different methods. Tell which you think is best and why.
In problem solve the IVP numerically on the suggested interval, if given using various step sizes. Compare with values of exact solutions when possible. y' = t2 + e-y, y(0) = 1; [0, 2]
For each of problem answer the following question a. Does picard's theorem apply to the given IVP Explain b. If your answer to part (a) is yes, is there a largest rectangle for which picard's conditions hold c. If you answer to part (a) is no are there other initial condition y(t0) = y0 for which
For Problems, answer the following questions for each of the points A, B, C, and D:(a) Does the differential equation seem to have a unique solution through the point?(b) If yes, on what interval do you think it is defined?1.2. 3.
We will investigate the initial-value problem y' = y2, y(0) = 1. (a) Show that Picard's conditions hold. How large a region R can be found? (b) Draw the direction field and solution. (c) Solve the IVP using separation of variables. What is the largest t-interval for which this solution is defined?
Show that the IVP y' = y1/3, y (0) = 0 exhibits non-unique solutions and sketch graphs of several possibilities. What does Picard's Theorem tell you for this problem?
For the JVP y' = √y. y (0) = 0 of Example 3, and for any positive number t0 show that a solution is given by equation (4).
Look back to Fig. 1.3.1, at the solution in the lower right. Follow it up and backward, toward y = -1. Does it actually merge with the solution y = - 1? Use Picard's Theorem to answer.
The conditions of Picard's Theorem may fail at a given point for a differential equation, but the equation may still have a unique solution through the point. (In other words, the converse of Picard's Theorem does not hold.) (a) Show that the uniqueness condition for Picard's Theorem does not apply
If you are given an empty bucket like the one in Fig. 1.5.8, having a hole in the bottom from which all the water has leaked, can you tell how long ago the bucket was full? Of course not, and the answer can be related to non-unique solutions. The equation describing the height h of the water level
The rate of change dV/dt of the volume V of a melting snowball is proportional to its surface area, sofor positive constant of proportionality k. (a) Explain how the relationship between the surface area and volume of a sphere leads to the power 2/3. (b) Suppose that you find a puddle of water that
Suppose that a spherical raindrop falling through a moist atmosphere grows at a rate proportional to its surface area. (a) Explain why dV/dt = kV2/3 k a positive constant, models this situation. (b) Demonstrate non-uniqueness for dV/dt = k V2/3, V(0) = 0, by constructing several solutions.
(a) Show that for arbitrary a ‰¥ 0, y' = y has infinitely many solutions that can be writteny(t) = e(t-a)(b) Show that for arbitrary a ‰¥ 0, the IVP s' = 2ˆšS, s(0) = 0, has infinitely many solutions(c) Sketch the similar graphs of the two families in (a) and (b).
In problems use the function given as the initial approximation y0 to the solution of the IVP y' = 1 - y, y(0) = I , And generate the first three Picard approximations y1, y2, y3, in each case. a. y0(t) = 1 b. y0(t) = t-1
Calculator or Computer Use Picard's Theorem and direction fields to study the following DEs (a) At what points do the corresponding NPs fail to have solutions? (b) At what points does uniqueness fail? 1. y' = y ¼ 2. y' = sin ty
The general (and basic) first-order linear differential equation y' + p(t)y = q(t), Where p a n d q ar e continuous functions on an interval l, will be studied in Sec. 2.2. Show that the Picard conditions apply to any linear DE initial-value problem y' + p (t) y = q(t). t(t0) = y0. for any to in
An operator is linear if it satisfies the two linear properties (3) and (4): otherwise, it is nonlinear. Which of the following differential operators are linear and which are nonlinear? a. L(y) = y' + 2y b. L(y) = y' + y2
Solve each of the following equations by inspection in less than 10 seconds. If the equation has an initial condition finds the arbitrary constant in the general solution. a. y' + 2y = 1 b. y' + y = 2 c. y' - 0.08y = 100 d. y'- 3y = 5
Show that if y1 (t) and y2 (t) are solutions of y' + p(t)y = 0 then so are y1 (t) + y2(t) and cy1 (t) for any constant c.
Show that if y1(t) and y2(t) are solutions of y" + p(t) y' +q(t)y = 0, then so is c1y1(t) + c2y2 (t) for any constants c1 and c2.
For Problems, verify that the given functions y1 and y2 are solutions of the given differential equation, then show that c1 y1 (t) + c2 y2 (t) is also a solution for any real number c1 and c2. a. y′′ − 9y = 0; y1 = e3t y2 = e-3t b. y′′ + 4y = 0; y1 = sin 2t y2 = cos2t
The function y(t) = t2 is a solution of y' - 2/t y = 0. Can you find any more solutions? Why do you know these other functions are solutions without substituting them into the equation?
For each of the non-homogeneous linear DEs in Problems (a) Verify that the given y1, y2, y3 satisfy the corresponding homogeneous equation. (b) Use the Superposition Principle, with appropriate coefficients, to state the general solution Yh (t) to the corresponding homogeneous equation. (c) Verify
Find general solutions for the equations given in Problems a. dy/dt + 2y = 0 b. dy/dt + 2y = 3et c. dy/dt - y = 3et
For Problems find solutions of the given IVPs. a. dy/dt - y = 1, y(0) = 1 b. dy/dt +2ty = t3, y(1) = 1
In Problems, solve each DE by the integrating factor method, Steps 1-4. a. y' + 2y = 0 b. y' + 2y = 3et
Solve the nonlinear IVP dy/dt = 1(t+y), y(-1) = 0 by reinterpreting it with y as the independent variable and t as the dependent variable.
The differential equation dy / dt = y2 / (ey - 2ty) looks impossible to solve analytically, but by treating y as the independent variable and t as the dependent variable, an implicit solution can be found. Carry out this solution.
(a) Use the change of variable z = In y to solve the nonlinear DE dy / dt + ay = by ln y, where a and bare constants. (b) Use the result from (a) to solve dy / dt + y = y ln y .
The nonlinear equation dy/dt +p(t) y = q(t) ya (where a ≠ 0, a ≠ 1) is called a Bernoulli equation and can be transformed into a linear equation. It already looks almost linear, except for ya on the right side. (a) Divide (19) by y" and then show that the transformation v = y1-a reduces (19) to
Solve the Bernoulli equation in problems. a. y' + ty = ty3 b. y' - y = ety2
For the linear DEs in Problem (a) Use an open-ended graphical DE solver to draw a direction field and some solutions. (b) Find the exact solution and relate it to your picture in (a). (c) If there is a steady-state solution, add it to your picture in (a) and highlight it in color. Then identify the
Consider the linear DEs in Problem with y(0) = 1 (a) Use a computer package with different numerical methods to find y(1) (b) Use the exact solution (as found in the corresponding problem to computer y(1) and compare with your approximations (c) Summarize what happens with different methods and
The direction fields of three DEs are shown in Fig. 2.2.3.(a) Label the equations linear or nonlinear. Of the two that are linear, which is homogeneous and which is non homogeneous?(b) Explain (algebraically) why the sum of two solutions of a homogeneous linear DE is a solution. Verify that the
From your conclusions in Problem, decide which of the direction fields in Fig. 2.2.4 represent linear homogeneous equations. Explain your reasons in each case
The time th required for the solution y of the decay problem y' = ky, k < 0, y(0) = y0 To reach one-half of its original value is called the half-life. (a) Find the half-life th in terms of the decay rate k. (b) Show that if the solution has value B at any time t1, then the solution will have the
If Q1 and Q1 are the amounts of a radioactively decaying substance at time t1 and t2 respectively, where t1
The U.S. government has dumped roughly· 100,000 barrels of radioactive waste into the Atlantic and Pacific oceans. The waste is mixed with concrete and encased in steel drums. The drums will eventually rust and seawater will gradually leach the radioactive material from the concrete and diffuse it
In 1964, Soviet scientists made a new element with atomic number 104, called simply E104, by bombarding plutonium with neon ions. The half-life of this new element is 0.15 seconds, and it was produced at a rate of 2 x 10-5 micrograms per second. Assum.ing none was present initially, how much E104
In many states it is illegal to drive with a blood alcohol level greater than 0.10% (one part alcohol per 1,000 parts blood) suppose someone who was involved in an automobile accident had blood alcohol tested at 0.20% at the time of the accident. Assume that the percentage of alcohol in the blood
In the tragic 1989 accident of the Exxon ship Valdez that dumped 240,000 barrels of oil into Prince William Sound, the National Safety Board determined that blood tests of Capt. Joseph Hazelwood showed a blood-alcohol content of 0.06%. 1 This testing did not take place until nine hours after the
Ed is undergoing surgery for an old football injury and must be anesthetized. The anesthesiologist knows Ed will be "under" when the concentration of sodium pentathol in his blood is at least 50 milligrams per kilogram of body weight. Suppose that Ed weighs 1 00 kg (220 pounds) and that sodium
The fact that sunlight is absorbed by water is well known to any diver who has dived to a depth of 100 ft. It is also true that the intensity of light falls exponentially with depth. Suppose that at a depth of 25 ft the water absorbs 15% of the light that strikes the surface. At what depth would
The number of bacteria in a colony increases at a rate proportional to the number present. If the number of bacteria doubles in 10 hours, how long will it take for the colony to triple in size?
If the number of bacteria in a culture is 5 million at the end of 6 hours and 8 million at the end of 9 hours, how many were present initially?
The number of bac1eria in a yeast culture grows at a rate proportional to the number present. If the population of a colony of yeast bacteria doubles in one hour, and if 5 million are present initially, find the number of bacteria in the colony after 4 hours.
A certain colony of bacteria grows at a rate proportional to the number of bacteria present. Suppose that the number of bacteria doubles every 12 hours. How long will it take this colony to grow to five times its original size?
A strain of tuberculosis bacteria grows at a rate proportional to its size. A researcher has determined that every hour the culture is 1.5 times larger than the hour before and that initially 1here were 100 cells present how many cells are present at any time t?
On an island that had no cats, the mouse population doubled during the first l0 years, reaching 50,000. At that time the islanders imported several cats that thereafter ate 6,000 mice every year. (a) What is the number of mice on the island t years after the arrival of the cats? (b) How many mice
A banker once gave the interpreta1ion of the constant e as the value after 10 years of an account earning 10% interest continuously compounded if the initial deposit is one dollar. Explain the merit of this claim.
In banking circles, the "Rule of 70" states that the time (in years) required for the value of an account to double in value can be approximated by dividing 70 by the annual interest rate (as a percentage, not a decimal). What is the reasorting behind this rule?
In l820, a William Record of London deposited $0.50 (or its equivalent in English pounds) for his granddaughter in the Bank of London. Unfortunately, he died before he could tell his granddaughter about the account. One hundred sixty years later, in 1980. The granddaughter's heirs discovered the
Upon entering college, Meena borrowed the limit of $5,000 on her credit card to help pay expenses. The credit company charges 19.95% annual interest, compounded continuously. How much will Meena owe when she graduates in four years?
In 1944 a Hollywood publicist decided to dramatize the opening of the movie Knickerbocker Holiday by arranging a stunt in which three bottles of whiskey originally thought to have been given to the Canarsie Indians for the island of Manhattan were to have been returned to the mayor of New York City
Sheryl's grandfather told Sheryl that 50 years ago the average cost of a new car was only $1.000, while today the average cost is S18.000. What continuous interest rate over the past 50 years would produce this change?
Upon graduating from college, Sergei has no money. However, during each year after that he will deposit d = $ 1,000 into an account that pays interest at a rate of 8% compounded continuously. (a) Find the future value A (t) of Sergei's account. (b) Find the value for an annual deposit d that
The reciprocal |I/k| (which has units of time) of the absolute value of the decay constant k in the decay equation y' = ky can be roughly interpreted as the time for y to fall two-thirds of the way from the initial value y0 to the limiting value 0. Show why this is true and illustrate with a figure.
Suppose a rich uncle has left you A0 dollars which is invested at rate r compounded continuously. Show that if you make withdrawals amounting to d dollars per year (where d > r A0) the time required to deplete the account to zero isWhat happens to the account when the annual withdrawal is not
Linda has won the New Jersey megabucks lottery consisting of one million dollars. Suppose that she deposits the money in a savings account that pays an annual rate of 8% compounded continuously. How long will the money last if she makes annual withdrawals of $100,000?
You must be careful about money. Lottery winners sometimes think they are millionaires when they're not really as rich as they think. Furthermore, there are enormous income taxes to be paid. But in these problems we are just calculating pretax earnings. Suppose that a state lottery's Grand Prize is
Many banks advertise that they compound interest continuously, meaning that the amount of money A (t) in an account satisfies the DE A' = r A, where r is the annual interest rate and t is time in years. (a) Show that an interest rate of8% compounded continuously gives an effective annual interest
After college you have no money, but you begin to create a retirement account by making continuous deposits that total d = $5, 000 per year. Suppose that the account pays interest at an annua1 rate of 8% compounded continuously. Use a computer or ca1culator to plot the future value of your account
Kelly and friends buy a house after graduating from college and borrow $200,000 from the bank to pay for it. Suppose that the bank charges 12% annual interest on the outstanding principle and that Kelly's group plans to make monthly payments of d = $2,500 to the bank. Call A(t) the amount of money
A certain radioactive material is known to decay at a rate proportional to the amount present. Over a 50-year period, an initial amount of 100 grams has decayed to only 75 grams. Find an expression for the amount of material r years after the initial measurement. Calculate the half-life of the
A certain radioactive substance has a half-life of 5 hours. Find the time for a given amount to decay to one-tenth of its original mass
Thorium-234 is a radioactive isotope that decays at a rate proportional to the amount present. Suppose that I gram of this material is reduced to 0.80 grams in one week. (a) Find an expression for the amount of Th-234 present at a general time t. (b) Find the half-life of Th-234. (c) Find the
A cypress beam found in the tomb of Sneferu in Egypt contained 55% of the amount of Carbon-14 found in living cypress wood. Estimate the age of the tomb.

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