What is an integrating factor and how does it help you solve a first-order linear differential equation?
> Show that the function f (t) = (e-t + 1)-1 satisfies y’ + y2 = y, y(0) = ½.
> Show that the function f (t) = 5 e2t satisfies y’’ - 3y’ + 2y = 0, y(0) = 5, y’ (0) = 10.
> Show that the function f (t) = t2 – ½ is a solution of the differential equation (y’ )2 - 4y = 2.
> Show that the function f (t) = 3/2 et2 – ½ is a solution of the differential equation y’ - 2ty = t.
> Solve the differential equation. y’ = y/t - 3y, t > 0
> Solve the differential equation. y ’ / (t + 1) = y + 1
> Solve the differential equation. y2y’ = 4t3 - 3t2 + 2
> Use Euler’s method with n = 5 on the interval 0 ≤ t ≤ 1 to approximate the solution f (t) to y ’ = ½ y (y - 10), y(0) = 9.
> Use Euler’s method with n = 6 on the interval 0 ≤ t ≤ 3 to approximate the solution f (t) to y ’ = .1 y(20 - y), y(0) = 2.
> Let f (t) be the solution to y ’ = (t + 1)/y, y(0) = 1. Use Euler’s method with n = 3 on 0 ≤ t ≤ 1 to estimate f (1). Then, show that Euler’s method gives the exact value of f (1) by solving the differential equation.
> The function f (t) = 5000/(1 + 49e-t) is the solution of the differential equation y’ = .0002y(5000 - y) from Example 8. (a) Graph the function in the window [0, 10] by [-750, 5750]. (b) In the home screen, compute .0002 f (3)(5000 - f
> Let f (t) be the solution to y ’ = 2e2t-y, y(0) = 0. Use Euler’s method with n = 4 on 0 ≤ t ≤ 2 to estimate f (2). Then show that Euler’s method gives the exact value of f (2) by solving the differential equation.
> A continuous annuity of $12,000 per year is to be funded by steady withdrawals from a savings account that earns 6% interest compounded continuously. (a) What is the smallest initial amount in the account that will fund such an annuity forever? (b) What
> A bank account has $20,000 earning 5% interest compounded continuously. A pensioner uses the account to pay himself an annuity, drawing continuously at a $2000 annual rate. How long will it take for the balance in the account to drop to zero?
> Suppose that in a chemical reaction, each gram of substance A combines with 3 grams of substance B to form 4 grams of substance C. The reaction begins with 10 grams of A, 15 grams of B, and 0 grams of C present. Let y = f (t) be the amount of C present a
> The birth rate in a certain city is 3.5% per year, and the death rate is 2% per year. Also, there is a net movement of population out of the city at a steady rate of 3000 people per year. Let N = f (t) be the city’s population at time t. (a) Write a diff
> Sketch the solution of the differential equation. Also indicate the constant solution. y’ = y3 - 6y2 + 9y, y(0) = - 1/4, y(0) = 1/4, y(0) = 4
> Sketch the solution of the differential equation. Also indicate the constant solution. y ’ = .4y2(1 - y), y(0) = -1, y(0) = .1, y(0) = 2
> Sketch the solution of the differential equation. Also indicate the constant solution. y’ = 3/(y + 3), y(0) = 2
> Sketch the solution of the differential equation. Also indicate the constant solution. y’ = 1/ (y2 + 1), y(0) = -1
> Sketch the solution of the differential equation. Also indicate the constant solution. y’ = 1 + cos y, y(0) = - 3/4
> Consider the differential equation y ’ = .2(10 - y) from Example 6. If the initial temperature of the steel rod is 510˚, the function f (t) = 10 + 500 e-0.2t is the solution of the differential equation. (a) Graph the function in the window [0, 30] by [-
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = ln y, y(0) = 2
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = y2 - 2y + 1, y(0) = -1
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = y2 + y, y(0) = - 13
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = 5 + 4y - y2, y(0) = 1
> Sketch the solution of the differential equation. Also indicate the constant solution. y' = 2 cos y, y(0) = 0
> Solve the initial-value problem y ’ = ey2 (cos y)(1 - ey-1), y(0) = 1.
> If f (t) is a solution of y = (2 - y) e-y, is f (t) increasing or decreasing at some value of t where f (t) = 3?
> Let P(t) denote the price in dollars of a certain commodity at time t in days. Suppose that the rate of change of P is proportional to the difference D - S of the demand D and supply S at any time t. Suppose further that the demand and supply are related
> Find a curve in the xy-plane passing through the origin and whose slope at the point (x, y) is x + y.
> Solve the differential equation. y – 1/2(1 + t) y = 1 + t, t ≥ 0
> Answer the question in Exercise 29 by using the differential equation to determine the sign of f ‘(t). Exercise 29: If y0 > 1, is the solution y = f (t) of the initial-value problem y’ = 2y (1 - y), y (0) = y0, dec
> Solve the differential equation. yy ’ + t = 6t2, y(0) = 7
> Solve the differential equation. y’ = 5 - 8y, y(0) = 1
> Solve the differential equation. yy ’ + t = 6t2, y(0) = 7
> Solve the differential equation. y’ = tet+y, y(0) = 0
> Solve the differential equation. y' = 7y ’ + ty’ , y(0) = 3
> Solve the differential equation. (y’ )2 = t
> What is a constant solution to a differential equation?
> What is a solution curve?
> What does it mean for a function to be a solution to a differential equation?
> What is a differential equation?
> If y0 > 1, is the solution y = f (t) of the initial-value problem y’ = 2y (1 - y), y (0) = y0, decreasing for all t > 0? Answer this question based on the slope field shown in Fig. 8. Figure 8: Y 3 2 1 0 0 1 2 3 4 t
> Describe Euler’s method for approximating the solution of a differential equation.
> What is the logistic differential equation?
> Outline the procedure for sketching a solution of an autonomous differential equation.
> How do you recognize an autonomous differential equation from its slope field?
> What is an autonomous differential equation?
> What is a first-order linear differential equation?
> Describe the separation-of-variables technique for obtaining the solution to a differential equation.
> What is the slope field?
> Suppose that f (t) satisfies the initial-value problem y = y2 + ty - 7, y(0) = 3. Is f (t) increasing or decreasing at t = 0?
> Figure 8 shows a portion of the solution curve of the differential equation y’ = 2y (1 - y) through the point (0, 2). On Fig. 8 or a copy of it, draw an approximation of the solution curve of the differential equation yâ€
> Find two constant solutions of y ’ = 4y (y - 7).
> Suppose that f (t) is a solution of y ’ = t2 - y2 and the graph of f (t) passes through the point (2, 3). Find the slope of the graph when t = 2.
> Suppose that f (t) is a solution of the differential equation y ’ = ty - 5 and the graph of f (t) passes through the point (2, 4). What is the slope of the graph at this point?
> The differential equation y’ = 2ty + et2, y(0) = 5, has solution y = (t + 5)et2. In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values calcula
> The differential equation y’ = et - 2y, y(0) = 1, has solution y = 1/3 (2e-2t + et ). In the following table, fill in the second row with the values obtained from the use of a numerical method and the third row with the actual values ca
> The differential equation y ’ = .5(y - 1)(4 - y) has five types of solutions labeled A–E. For each of the following initial values, graph the solution of the differential equation and identify the type of solution. Use a small value of h, let t range fro
> The differential equation y ’ = .5(1 - y)(4 - y) has five types of solutions labeled A–E. For each of the following initial values, graph the solution of the differential equation and identify the type of solution. Use a small value of h, let t range fro
> The Los Angeles Zoo plans to transport a California sea lion to the San Diego Zoo. The animal will be wrapped in a wet blanket during the trip. At any time t, the blanket will lose water (due to evaporation) at a rate proportional to the amount f (t) of
> Suppose that the Consumer Products Safety Commission issues new regulations that affect the toy-manufacturing industry. Every toy manufacturer will have to make certain changes in its manufacturing process. Let f (t) be the fraction of manufacturers that
> Let f (t) be the solution of y ’ = 10 - y, y(0) = 1. Use Euler’s method with n = 5 to estimate f (1). Then, solve the differential equation and find the exact value of f (1).
> Let f (t) be the solution of y’ = -(t + 1)y2, y(0) = 1. Use Euler’s method with n = 5 to estimate f (1). Then, solve the differential equation, find an explicit formula for f (t), and compute f (1). How accurate is the estimated value of f (1)?
> Figure 8 shows a slope field of the differential equation y’ = 2y (1 - y). With the help of this figure, determine the constant solutions, if any, of the differential equation. Verify your answer by substituting back into the equation.
> Let f (t) be the solution of y’ = y(2t - 1), y(0) = 8. Use Euler’s method with n = 4 to estimate f (1).
> Use Euler’s method with n = 4 to approximate the solution f (t) to y’ = 2t - y + 1, y(0) = 5 for 0 ≤ t ≤ 2. Estimate f (2).
> Use Euler’s method with n = 2 on the interval 2 ≤ t ≤ 3 to approximate the solution f (t) to y’ = t - 2y, y(2) = 3. Estimate f (3).
> Use Euler’s method with n = 2 on the interval 0 ≤ t ≤ 1 to approximate the solution f (t) to y = t2y, y(0) = -2. In particular, estimate f (1).
> Suppose that f (t) satisfies the initial-value problem y = y2 + ty - 7, y(0) = 2. Is the graph of f (t) increasing or decreasing at t = 0?
> You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed l
> You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed l
> In the study of the effect of natural selection on a population, we encounter the differential equation dq/dt = -.0001q2 (1 - q), where q is the frequency of a gene a and the selection pressure is against the recessive genotype aa. Sketch a solution of
> Show that the mathematical model in Check Your Understanding 2 predicts that the amount of litter in the forest will eventually stabilize. What is the “equilibrium level” of litter in that problem? [Note: Today, most forests are close to their equilibriu
> A single dose of iodine is injected intravenously into a patient. The iodine mixes thoroughly in the blood before any is lost as a result of metabolic processes (ignore the time required for this mixing process). Iodine will leave the blood and enter the
> On the slope field in Fig. 4(a) or a copy of it, draw an approximation of a portion of the solution curve of the differential equation y’ = t - y that goes through the point (0, 2). In your opinion, based on the slope field, can this solution pass throu
> A certain drug is administered intravenously to a patient at the continuous rate of 5 milligrams per hour. The patient’s body removes the drug from the bloodstream at a rate proportional to the amount of the drug in the blood. Write a differential equati
> The air in a crowded room full of people contains .25% carbon dioxide (CO2). An air conditioner is turned on that blows fresh air into the room at the rate of 500 cubic feet per minute. The fresh air mixes with the stale air, and the mixture leaves the r
> A company arranges to make continuous deposits into a savings account at the rate of P dollars per year. The savings account earns 5% interest compounded continuously. Find the approximate value of P that will make the savings account balance amount to $
> A company wishes to set aside funds for future expansion and so arranges to make continuous deposits into a savings account at the rate of $10,000 per year. The savings account earns 5% interest compounded continuously. (a) Set up the differential equat
> A continuous annuity is a steady stream of money that is paid to some person. Such an annuity may be established, for example, by making an initial deposit in a savings account and then making steady withdrawals to pay the continuous annuity. Suppose tha
> The fish population in a pond with carrying capacity 1000 is modeled by the logistic equation dN/dt = .4 1000 N(1000 - N ). Here, N(t) denotes the number of fish at time t in years. When the number of fish reached 275, the owner of the pond decided t
> Consider a certain commodity that is produced by many companies and purchased by many other firms. Over a relatively short period, there tends to be an equilibrium price p0 per unit of the commodity that balances the supply and the demand. Suppose that,
> In economic theory, the following model is used to describe a possible capital investment policy. Let f (t) represent the total invested capital of a company at time t. Additional capital is invested whenever f (t) is below a certain equilibrium value E,
> L. F. Richardson proposed the following model to describe the spread of war fever. If y = f (t) is the percentage of the population advocating war at time t, the rate of change of f (t) at any time is proportional to the product of the percentage of the
> Suppose that substance A is converted into substance B at a rate that, at any time t, is proportional to the square of the amount of A. This situation occurs, for instance, when it is necessary for two molecules of A to collide to create one molecule of
> The health officials that studied the flu epidemic in Example 8 made an error in counting the initial number of infected people. They are now claiming that f (t) (the number of infected people after t days) is a solution of the initial-value problem y’
> An experimenter reports that a certain strain of bacteria grows at a rate proportional to the square of the size of the population. Set up a differential equation that describes the growth of the population. Sketch a solution.
> Let c be the concentration of a solute outside a cell that we assume to be constant throughout the process, that is, unaffected by the small influx of the solute across the membrane due to a difference in concentration. The rate of change of the concentr
> A porous material dries outdoors at a rate that is proportional to the moisture content. Set up the differential equation whose solution is y = f (t), the amount of water at time t in a towel on a clothesline. Sketch the solution.
> In an autocatalytic reaction, one substance is converted into a second substance in such a way that the second substance catalyzes its own formation. This is the process by which trypsinogen is converted into the enzyme trypsin. The reaction starts only
> At one point in his study of a falling body starting from rest, Galileo conjectured that its velocity at any time is proportional to the distance it has dropped. Using this hypothesis, set up the differential equation whose solution is y = f (t), the dis
> For information being spread by mass media, rather than through individual contact, the rate of spread of the information at any time is proportional to the percentage of the population not having the information at that time. Give the differential equat
> Answer parts (a) and (b) in Example 2 if the pond has a carrying capacity of 2000 fish and all other data are unchanged. Example 2: A pond on a fish farm has a carrying capacity of 1000 fish. The pond was originally stocked with 100 fish. Let N(t) denot
> Answer part (a) in Example 2 if the pond was originally stocked with 600 fish and all other data are unchanged, how does the graph of the fish population in this case differ from the one in Example 2? Example 2: A pond on a fish farm has a carrying capa
> You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed l
> You are given a logistic equation with one or more initial conditions. (a) Determine the carrying capacity and intrinsic rate. (b) Sketch the graph of dN/dt versus N in an Nz-plane. (c) In the tN-plane, plot the constant solutions and place a dashed l
> On the slope field in Fig. 5(a) or a copy of it, draw the solution of the initial-value problem y’ = .0002y (5000 - y), y(0) = 500. Figure 5: 5000 4000 3000 2000 1000 0 0 1 2 3 4 5 6 t (a) Slope field of y = .0002y(5000 - y). y In
> Sketch the graph of a function with the stated properties. Domain: 0 ≤ t ≤ 4; (0, 2) is on the graph; the slope is always positive, and the slope becomes more positive (as t increases).
