A bank account earns 5% annual interest compounded continuously. Continuous payments are

Match the graphs in Figure 9.48 with the following descriptions. (a) The population of a new species introduced onto a tropical island. (b) The temperature of a metal ingot placed in a furnace and then removed. (c) The speed of a car traveling at uniform speed and then braking uniformly. (d) The mass of carbon-14 in a historical specimen. (e) The concentration of tree pollen in the air over the course of a year. t (I) t (II) t (III) t (IV) t (V) Figure 9.48

(a) Determine which of the following functions is a solution to the differential equation x dy dx = 3y. (i) y = Cx2 (ii) y = Cx3 (iii) y = x3 + C (b) For any function which is a solution, find C if y = 40 when x = 2.

Is y = x3 a solution to the differential equation xy 3y = 0? Justify your answer.

For a certain quantity y, assume that dy/dt = 0.20y. Fill in the values of y in Table 9.7. Assume that the rate of growth given by dy/dt is approximately constant over each unit time interval. Table 9.7 t 0 1 2 3 4 y 125

Slope fields for dy/dx = 1+x and dy/dx = 1+y are in Figures 9.49 and 9.50. (a) Which slope field corresponds to which equation? (b) On each slope field, draw the solution curve through the origin. (c) For each slope field, list all equilibrium solutions and indicate whether each is stable or unstable. 2 1 1 2 2 1 1 2 x (I) y Figure 9.49 2 1 1 2 2 1 1 2 x (II) y Figure 9.50

(a) Sketch the slope field for the equation y = xy in Figure 9.51 at the points indicated. (b) Check that y = x 1 is the solution to the differential equation passing through the point (1, 0). 2 1 1 2 2 1 1 2 x y Figure 9.51

Which one of the following differential equations best fits the slope field shown in Figure 9.52? Explain. I. y = 1+y II. y = 2 y III. y = (1+y)(2 y) IV. y = 1+x V. y = xy

Match each of the slope field segments in (I)(VI) with one or more of the differential equations in (a)(f). (a) y = ex2 (b) y = cos y (c) y = cos(4 y) (d) y = y(4 y) (e) y = y(3 y) (f) y = x(3 x) 4 4 x (I) y 4 4 x (II) y 4 4 x (III) y 4 4 x (IV) y 4 4 x (V) y 4 4 x (VI) y

A deposit is made to a bank account paying an annual interest rate of 7% compounded continuously. No other deposits or withdrawals are made to the account. (a) Write a differential equation satisfied by B, the balance in the account after t years. (b) Solve the differential equation given in part (a). (c) If the initial deposit is $5000, give the particular solution satisfying this initial condition. (d) How much is in the account after 10 years?

Radioactive iodine decays at a continuous rate of about 9% per day. Write a differential equation to model this behavior. Find the general solution.

Solve the differential equations in Problems 1120. dP dt = t

Solve the differential equations in Problems 1120. dy dt = 5y

Solve the differential equations in Problems 1120. dy dt = 5t

Solve the differential equations in Problems 1120. dP dt = 0.03P

Solve the differential equations in Problems 1120. dA dt = 0.07A

Solve the differential equations in Problems 1120. 1 Q dQ dt = 2

Solve the differential equations in Problems 1120. dP dt = 10 2P

Solve the differential equations in Problems 1120. dy dt = 100 y

Solve the differential equations in Problems 1120. dy dx = 0.2y 8

Solve the differential equations in Problems 1120. dH dt = 0.5H + 10

Write a differential equation whose solution is the temperature as a function of time of a bottle of orange juice taken out of a 40F refrigerator and left in a 65F room. Solve the equation and graph the solution.

The rate of growth of a tumor is proportional to the size of the tumor. (a) Write a differential equation satisfied by S, the size of the tumor, in mm, as a function of time, t. (b) Find the general solution to the differential equation. (c) If the tumor is 5 mm at time t = 0, what does that tell you about the solution? (d) If, in addition, the tumor is 8 mm at time t = 3, what does that tell you about the solution?

The radioactive isotope carbon-14 is present in small quantities in all life forms, and it is constantly replenished until the organism dies, after which it decays to stable carbon-12 at a rate proportional to the amount of carbon-14 present, with a half-life of 5730 years. Suppose C(t) is the amount of carbon-14 present at time t. (a) Find the value of the constant k in the differential equation C = kC. (b) In 1988 three teams of scientists found that the Shroud of Turin, which was reputed to be the burial cloth of Jesus, contained 91% of the amount of carbon-14 contained in freshly made cloth of the same material.10 How old is the Shroud of Turin, according to these data?

A bank account earns 5% annual interest compounded continuously. Continuous payments are made out of the account at a rate of $12,000 per year for 20 years. (a) Write a differential equation describing the balance B = f(t), where t is in years. (b) Solve the differential equation given an initial balance of B0. (c) What should the initial balance be such that the account has zero balance after precisely 20 years?

Warfarin is a drug used as an anticoagulant. After administration of the drug is stopped, the quantity remaining in a patients body decreases at a rate proportional to the quantity remaining. The half-life of warfarin in the body is 37 hours. (a) Sketch the quantity, Q, of warfarin in a patients body as a function of the time, t, since stopping administration of the drug. Mark the 37 hours on your graph. (b) Write a differential equation satisfied by Q. (c) How many days does it take for the drug level in the body to be reduced to 25% of the original level?

Dead leaves accumulate on the ground in a forest at a rate of 3 grams per square centimeter per year. At the same time, these leaves decompose at a continuous rate of 75% per year.Write a differential equation for the total quantity of dead leaves (per square centimeter) at time t. Sketch a solution showing that the quantity of dead leaves tends toward an equilibrium level. What is that equilibrium level?

Morphine is often used as a pain-relieving drug. The halflife of morphine in the body is 2 hours. Suppose morphine is administered to a patient intravenously at a rate of 2.5 mg per hour, and the rate at which the morphine is eliminated is proportional to the amount present. (a) Use the half-life to show that, to three decimal places, the constant of proportionality for the rate at which morphine leaves the body (in mg/hour) is k = 0.347. (b) Write a differential equation for the quantity, Q, of morphine in the blood after t hours. (c) Use the differential equation to find the equilibrium solution. (This is the long-term amount of morphine in the body, once the system has stabilized.)

(a) Find all equilibrium solutions for the equation dy dx = 0.5y(y 4)(2 + y). (b) With a calculator or computer, draw a slope field for this differential equation. Use it to determine whether each equilibrium solution is stable or unstable.

According to a simple physiological model, an athletic adult male needs 20 Calories per day per pound of body weight to maintain his weight. If he consumes more or fewer Calories than those required to maintain his weight, his weight changes at a rate proportional to the difference between the number of Calories consumed and the number needed to maintain his current weight; the constant of proportionality is 1/3500 pounds per Calorie. Suppose that a particular person has a constant caloric intake of I Calories per day. Let W(t) be the persons weight in pounds at time t (measured in days). (a) What differential equation has solution W(t)? (b) Find the equilibrium solution of the differential equation. Based on the context, do you expect the equilibrium to be stable or unstable? (c) Solve this differential equation. (d) Graph W(t) if the person starts out weighing 160 pounds and consumes 3000 Calories a day.

(a) Graph f(y) = y y2. (b) Sketch the slope field of the differential equation dy dx = y y2. (c) Describe how the two graphs are related. In particular, explain how you can determine the equilibrium solutions to the differential equation from the graph of f(y) and how you can tell if the equilibrium solutions are stable.

(a) What are the equilibrium solutions to the differential equation y = f(y), where f(y) is graphed in Figure 9.53? (b) Sketch the slope field for y = f(y). (c) For each of the following initial conditions, sketch the solution curve on your slope field: (i) y(0) = 0 (ii) y(0) = 1 (iii) y(0) = 6 (iv) y(0) = 8 (v) y(0) = 10 (vi) y(0) = 16 (vii) y(0) = 17 (d) Which of the equilibrium solutions are stable? Which are unstable?

Solve the differential equations in Problems 1120. The general solution of the differential equation dQ/dt = 0.25Q is Q = Ce0.25t.

Solve the differential equations in Problems 1120. The general solution of the differential equation dw/dr = 0.3r is w = Ce0.3r.

Solve the differential equations in Problems 1120. The function H = 57e0.5t is one solution to the differential equation dH/dt = 0.5t.

Solve the differential equations in Problems 1120. There is exactly one solution of the differential equation dy/dt = 3y which has y(0) = 5.

Solve the differential equations in Problems 1120. The particular solution of dy/dt = 2y satisfying y = 3 when t = 0 is y = Ce2t.

Solve the differential equations in Problems 1120. The balance B in a bank account that earns interest continuously at a rate of 3% per year on an initial deposit of $5000 has differential equation dB/dt = 5000B.

Solve the differential equations in Problems 1120. IfQ(t) is the quantity of a drug in a patients bloodstream at time t, and the rate at which the drug leaves the bloodstream is proportional to the quantity in the bloodstream, then dQ/dt = kQ where k is some negative constant.

Solve the differential equations in Problems 1120. A quantity, Q, of dead leaves decomposes at a rate proportional to the quantity present. We can model this with the differential equation Q = Cekt, where k is negative and t represents time.

Solve the differential equations in Problems 1120. A quantity, Q, of dead leaves decomposes at a rate proportional to the quantity present. We can model this with the differential equation dQ/dt = kQ, where k is negative and t represents time.

Solve the differential equations in Problems 1120. The general solution to the differential equation dy/dt = 0.3(y 75) is y = 75+Ce0.3t

Solve the differential equations in Problems 1120. The general solution to the differential equation dP/dt = 2P 100 is P = 100+Ce2t.

Solve the differential equations in Problems 1120. The general solution to the differential equation dQ/dt = 0.5(Q + 20) is Q = 20+Ce0.5t.

Solve the differential equations in Problems 1120. The general solution to the differential equation dW/dt = 600 3W isW = 200+Ce3t.

Solve the differential equations in Problems 1120. The solution to the differential equation dA/dt = 0.25(A 40) with initial condition A(0) = 50 is A = 40 + 50e0.25t .

Solve the differential equations in Problems 1120. The solution to the differential equation dQ/dt = 50 2Q with initial condition Q(0) = 100 is Q = 25 + 75e2t.

Solve the differential equations in Problems 1120. The balance, B, in a bank account earns 4% interest per year, compounded continuously, and payments are made out of the account at a rate of $12,000 per year. If t is in years, the differential equation to model this situation is B = 12,000 + Ce0.04t.

Solve the differential equations in Problems 1120. A drug enters a patients body intravenously at a continuous rate of 12 mg per hour, and the drug is metabolized at a continuous rate of 18%per hour. If A gives the quantity of drug in the body at hour t, then the differential equation to model this situation is dA/dt = 12 0.18A.

Solve the differential equations in Problems 1120. An equilibrium solution for the differential equation dH/dt = 0.037(H 225) is H = 225.

Solve the differential equations in Problems 1120. The graph of an equilibrium solution for a differential equation is a horizontal line.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. Assume dx/dt = ax+bxy and dy/dt = cy+dxy. If X would do fine if Y did not exist, then b must be positive.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. Assume dx/dt = ax + bxy and dy/dt = cy + dxy. If species X eats species Y , then c must be negative.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. Assume dx/dt = ax + bxy and dy/dt = cy + dxy. If species X eats species Y , then d must be negative.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. Assume dx/dt = ax + bxy and dy/dt = cy + dxy. If species X cannot survive without species Y and does fine with species Y , then a must be negative and b must be positive.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. If dx/dt = 0.02x 0.15xy and dy/dt = 0.05y 0.18xy, then both populations need the other to survive.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. If dx/dt = 0.02x 0.15xy and dy/dt = 0.05y 0.18xy, then both populations would do fine if the other population did not exist.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. If dx/dt = 0.02x 0.15xy and dy/dt = 0.05y 0.18xy, then the two populations have a negative impact on each other.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. If dx/dt = 0.12x + 0.07xy and dy/dt = 0.10y + 0.25xy, then the two populations need each other to survive..

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. Assume dx/dt = 3x + xy and dy/dt = 2y 5xy. If x = 1 and y = 2, then both populations will decrease in size.

In Problems 5160, indicate whether the statement is true or false. Assume x and y represent the sizes of populations of two interacting species X and Y with system of differential equations as given. Assume dx/dt = 3x + xy and dy/dt = 2y 5xy. If x = 2 and y = 5, then both populations will decrease in size.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have The derivative dS/dt is negative because people are dying from the epidemic.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have The parameter a in dS/dt = aSI is the same as the parameter a in dI/dt = aSI bI because people are moving from one group to the other.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have The term bI in dI/dt = aSI bI is negative because people are moving back to the susceptible group.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have The term bI in dI/dt = aSI bI is negative because people are either recovering or dying or being separated from the rest of the population and are no longer in either the S group or the I group.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have If Type I flu is much more contagious than Type II flu, then the parameter b will be larger for Type I flu.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have If Type I flu is much more contagious than Type II flu, then the parameter a will be larger for Type I flu.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have If Type I flu lasts 10 days and Type II flu lasts 3 days, then the parameter b will be larger for Type I flu.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have If Type I flu lasts 10 days and Type II flu lasts 3 days, then the parameter b will be smaller for Type I flu.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have If a = 0.001 and b = 0.3 and if current numbers are S = 500 and I = 100, then the number of sick people will increase.

In Problems 6170, indicate whether the statement is true or false. The problems refer to the SI R model, in which the variable S represents the number of people susceptible to becoming sick, the variable I represents the number of people currently infected, and R represents the rest of the population. We have If a = 0.001 and b = 0.3 and if current numbers are S = 100 and I = 500, then the number of sick people will increase.