Chapter 31

Solve the following differential equations by using the method of substitution to put them into the form \(\frac{d y}{d t}=k y\). (a) \(\frac{d P}{d t}=0.3(1000-P)\) (b) \(\frac{d M}{d t}=0.4 M-2000\)

Sketch a representative family of solutions for each of the following differential equations. (a) \(\frac{d y}{d t}=2 y-6\) (b) \(\frac{d y}{d t}=6-2 y\)

Find the particular solution corresponding to the initial conditions given. \(4 \frac{d^{2} x}{d t^{2}}-4 \frac{d x}{d t}=-x, \quad x(0)=1, \quad x^{\prime}(0)=2\)

Solve the following. (a) \(\frac{d x}{d t}=6-2 x\). Do this using substitution in two ways. i. Factor out a \(-2\) from the right-hand side and let \(u=x-3\). Then solve. ii. Let \(v=6-2 x .\) Express \(\frac{d x}{d t}\) in terms of \(v\) and then convert the equation \(\frac{d x}{d t}=6-2 x\) to an equation in \(v\) and \(\frac{d v}{d t}\) and solve. (b) \(\frac{d x}{d t}=3 x-7\) (c) \(\frac{d y}{d t}=k y+B\)

Sketch a representative family of solutions for each of the following differential equations. (a) \(\frac{d y}{d t}=\sin t\) (b) \(\frac{d y}{d t}=\sin y\)

Refer to the equation $$x^{\prime \prime}+b x^{\prime}+c x=0 .$$ What condition(s) must be satisfied to have a periodic solution? If the solution is periodic, what will its period be?

A very large container of juice contains four gallons of apple juice and one gallon of cranberry juice. Cranberry-apple juice \((60 \%\) apple, \(40 \%\) cranberry) is entering the container at a rate of three gallons per hour. The well-stirred mixture is leaving the container at three gallons per hour. (a) Write a differential equation whose solution is \(C(t)\), the number of gallons of cranberry juice in the container at time \(t .\) Solve, using the initial condition. (b) Write a differential equation whose solution is \(A(t)\), the number of gallons of apple juice in the container at time \(t .\) Solve the equation.

When a population has unlimited resources and is free from disease and strife, the rate at which the population grows in often modeled as being proportional to the population. Assume that both the bee and the mosquito populations described below behave according to this model. In both scenarios described below you are given enough information to nd the proportionality constant \(k\). In one case the information allows you to nd \(k\) solely using the differential equation, without requiring that you solve it. In the other scenario you must actually solve the differential equation in order to nd \(k\). (a) Let \(M=M(t)\) be the mosquito population at time \(t, t\) in weeks. At \(t=0\) there are 1000 mosquitoes. Suppose that when there are 5000 mosquitoes the population is growing at a rate of 250 mosquitoes per week. Write a differential equation re ecting the situation. Include a value for \(k\), the proportionality constant. (b) Let \(B=B(t)\) be the bee population at time \(t, t\) in weeks. At \(t=0\) there are 600 bees. When \(t=10\) there are 800 bees. Write a differential equation re ecting the situation. Include a value for \(k\), the proportionality constant.

Sketch a representative family of solutions for each of the following differential equations. \(\frac{d y}{d t}=\tan y\)

The population in a certain country grows at a rate proportional to the population at time \(t\), with a proportionality constant of \(0.03 .\) Due to political turmoil, people are leaving the country at a constant rate of 6000 people per year. Assume that there is no immigration into the country. Let \(P=P(t)\) denote the population at time \(t\). (a) Write a differential equation re ecting the situation. (b) Solve the differential equation for \(P(t)\) given the information that at time \(t=0\) there are 3 million people in the country. In other words, nd \(P(t)\), the number of people in the country at time \(t\).

Sketch a representative family of solutions for each of the following differential equations. (a) \(\frac{d y}{d t}=t^{2}\) (b) \(\frac{d y}{d t}=y^{2}\)

Refer to the equation $$x^{\prime \prime}+b x^{\prime}+c x=0 .$$ Suppose \(b>0\) and \(c>0 .\) Is \(\lim _{l \rightarrow \infty} x(t)\) necessarily zero? Explain. Interpret your response in terms of vibrating springs.

A population of otters is declining. New otters are born at a rate proportional to the population with constant of proportionality \(0.04\), but otters die at a rate proportional to the population with constant \(0.09\). Today, the population is 1000 . A group of people wants to try to prevent the otter population from dying out, so they plan to bring in otters from elsewhere at a rate of 40 otters per year. We ll model the situation with continuous functions. Let \(P(t)\) be the population of the otters \(t\) years after today. (a) Write a differential equation whose solution is \(P(t)\). (b) Solve this differential equation. Your answer should include no unknown constants. (c) According to this model, will the attempt to save the otter population work? Explain your answer. If it won \(\mathrm{t}\) work, at what rate must otters be brought in to ensure the population s survival? If it will work, for how many years must the importation of otters continue?

Sketch a representative family of solutions for each of the following differential equations. (a) \(\frac{d y}{d t}=t^{2}-1\) (b) \(\frac{d y}{d t}=y^{2}-1\)

Refer to the equation $$x^{\prime \prime}+b x^{\prime}+c x=0 .$$ Suppose \(b<0\) and \(c>0 .\) For \(x(0)\) and \(x^{\prime}(0)\) not both zero, is it possible that \(\lim _{t \rightarrow \infty}|x(t)|=L\), where \(L\) is finite? Explain.

Sketch a representative family of solutions for each of the following differential equations. $$ \frac{d x}{d t}=(1-x)(x+2)(x-3) $$

A spring with a \(2-\mathrm{kg}\) mass has a natural length of \(0.6 \mathrm{~m} .\) A \(10 \mathrm{~N}\) force is required to compress it to a length of \(0.5 \mathrm{~m}\). If the spring is compressed to \(0.4 \mathrm{~m}\) and released, find the position of the mass at time \(t\). Assume a frictionless system.

Money in a bank account is earning interest at a nominal rate of \(4 \%\) per year compounded continuously. Withdrawals are made at a rate of \(\$ 8000\) per year. Assume that withdrawals are made continuously. (a) Write a differential equation modeling the situation. (b) Depending on the initial deposit, the amount of money in the account will either increase, decrease, or remain constant. Explain this in words; refer to the differential equation. (c) Suppose the money in the account remains constant. What was the initial deposit? For what initial deposits will the amount of money in the account actually continue to grow? (d) Show that \(M(t)=M_{0} e^{0.04 t}-8000 t\) is not a solution to the differential equation you got in part (a).

Give an example of a differential equation with constant solutions at \(y=-1\) and \(y=4\) with the characteristics specified. (a) The equilibrium at \(y=-1\) is stable; the equilibrium at \(y=4\) is unstable. (b) The equilibrium at \(y=-1\) is unstable; the equilibrium at \(y=4\) is stable. (c) Neither equilibrium solution is stable.

Write a second order homogeneous differential equation that is satisfied by \(y(t)=\) \(e^{t} \sin t .\) (The answer is not unique.)

Suppose a population is changing according to the equation \(\frac{d P}{d t}=k P-E\), where \(E\) is the rate at which people are emigrating from the country. As established in part (d) of the previous problem, \(P(t)=P_{0} e^{k t}-E t\) is not a solution to this differential equation. (a) Use substitution to solve \(\frac{d P}{d t}=k P-E\). (Your answer ought to agree with that given in part (b).) (b) Verify that \(P(t)=C e^{k t}+\frac{E}{k}\), where \(C\) is a constant, is a solution to the differential equation \(\frac{d P}{d t}=k P-E\)

Give an example of a differential equation of the form \(\frac{d v}{d t}=f(v)\) and whose solutions depend upon \(v(0)\) as described below. If \(v(0)>5\), then \(v(t)\) is increasing. If \(v(0)=5\), then \(v(t)=5\). If \(2

Compute the following. (a) \(e^{2 \pi i}\) (b) \(e^{-\pi i}\)

\(P(t)=C e^{k t}+\frac{E}{k}\), where \(C\) is a constant, is the general solution to the differential equation \(\frac{d P}{d t}=k P-E .\) Below is the slope eld for \(\frac{d P}{d t}=2 P-6\). (a) i. Find the particular solution that corresponds to the initial condition \(P(0)=2\). ii. Sketch the solution curve through \((0,2)\). (b) i. Find the particular solution that corresponds to the initial condition \(P(0)=3\). ii. Sketch the solution curve through \((0,3)\). (c) i. Find the particular solution that corresponds to the initial condition \(P(0)=4\). ii. Sketch the solution curve through \((0,4)\).

A canister contains 10 liters of blue paint. Paint is being used at a rate of 2 liters per hour and the canister is being replenished at a rate of 2 liters per hour by a pale blue paint that is \(80 \%\) blue and \(20 \%\) white. Assuming the canister is well-mixed, write a differential equation whose solution is \(w(t)\), the amount of white paint in the canister at time \(t\). Specify the initial condition.

A 5-gallon urn is filled with chai, a milky spicy tea. The chai in the urn is \(90 \%\) tea and \(10 \%\) milk. Chai is being consumed at a rate of \(1 / 2\) gallon per hour and the urn is kept full by adding a mixture that is \(80 \%\) tea and \(20 \%\) milk. Assume that the chai is well-mixed. (a) Write a differential equation whose solution is \(M(t)\), the number of gallons of milk in the urn at time \(t\). Specify the initial condition. (b) Use qualitative analysis to sketch the solution to the differential equation in part (a). (c) How much milk is in the urn after 2 hours?

A miser spends money at a rate proportional to the amount he has. Suppose that right now he has \(\$ 100,000\) stowed under his mattress; he does not pay any taxes and does not earn any return on his money. Assume that this is all the money he has and that he has no other source of income. At the moment he is spending the money at a rate of \(\$ 10,000\) per year. (a) At what rate will he be spending money when he has \(\$ 50,000 ?\) (b) At what time will the amount of money be down to \(\$ 10,000\) ?

The population of wildebeest in the Serengeti was decimated by a rinderpest plague in the \(1950 \mathrm{~s}\). In 1961 the Serengeti supported a population of a quarter of a million wildebeest. By 1978 the wildebeest population was \(1.5\) million and by 1991 it had reached 2 million. (Craig Packer Into Africa, Chicago, The University of Chicago Press, 1996 p. \(250 .\) ) Given this data, would you be more inclined to model the growth of the wildebeest population using an exponential growth model or using a logistic growth model? Explain your reasoning.

A drosophila colony (a colony of fruit ies) is being kept in a laboratory for study. It is being provided with essentially unlimited resources, so if left to grow, the colony will grow at a rate proportional to its size. If we let \(N(t)\) be the number of drosophila in the colony at time \(t, t\) given in weeks, then the proportionality constant is \(k\). (a) Write a differential equation re ecting the situation. (b) Solve the differential equation using \(N_{0}\) to represent \(N(0)\). (c) Suppose the drosophila are being cultivated to provide a source for genetic study, and therefore drosophila are being siphoned off at a rate of \(S\) drosophila per week. Modify the differential equation given in part (a) to re ect the siphoning off. (d) One of your classmates is convinced that the solution to the differential equation in part (c) is given by $$ N(t)=N_{0} e^{k t}-S t $$ Show him that this is not a solution to the differential equation. (e) Your classmate is having a hard time giving up the solution he brought up in part (d). He sees that it does not satisfy the differential equation, but he still has a strong gut feeling that it ought to be right. Convince him that it is wrong by using a more intuitive argument. Use words and talk about fruit ies.

A lake contains \(10^{10}\) liters of water. Acid rain containing \(0.02\) milligrams of pollutant per liter of rain falls into the lake at a rate of \(10^{3}\) liters per week. An outlet stream drains away \(10^{3}\) liters of water per week. Assume that the pollutant is always evenly distributed throughout the lake, so the runoff into the stream has the same concentration of pollutant as the lake as a whole. The volume of the lake stays constant at \(10^{10}\) liters because the water lost from the runoff balances exactly the water gained from the rain. (a) Write a differential equation whose solution is \(P(t)\), the number of milligrams of pollutant in the lake as a function of \(t\) measured in weeks. (b) Find any equilibrium solutions. (c) Sketch some representative solution curves. (d) How would you alter the differential equation if there was a dry spell and rain was falling into the lake at a rate of only \(10^{2}\) liters per week.

Solve the differential equations below. Find the general solution. (a) \(\frac{d y}{d t}=\sin 3 t\) (b) \(\frac{d y}{d t}=5 \cdot 2^{t}\) (c) \(\frac{d x}{d t}=\frac{t+1}{l}\) (d) \(\frac{d x}{d t}=\frac{t+1}{t^{2}}\)

(a) You plan to save money starting today at a rate of \(\$ 4000\) per year over the next 30 years. You will deposit this money at a nearly continuous rate (a constant amount each day) into a bank account that earns \(5 \%\) interest compounded continuously. Let \(B(t)\) be the balance of money in the account \(t\) years from now, where \(0 \leq t \leq 30\) i. Write a differential equation whose solution is \(B(t)\). ii. Write an integral that is equal to \(B(30)\), the amount in the account at the end of 30 years. (b) Now assume that instead of making deposits continuously, you decide to make a deposit of \(\$ 4000\) once a year, starting today and continuing until you have made a total of 30 deposits. Suppose the bank account pays \(5 \%\) interest compounded annually. i. Write a geometric sum equal to the balance immediately after the final deposit. ii. Find a closed form expression (no \(+\cdots+\), no summation notation) for this sum.

Solve the differential equations below. Find the general solution. (a) \(\frac{d y}{d t}=3 t+5\) (b) \(\frac{d y}{d t}=3 y\) (c) \(\frac{d y}{d t}=-y\) (d) \(\frac{d y}{d t}=0\) (e) \(\frac{d y}{d t}=3 y-6\)

For each differential equation below, sketch the slope eld and nd the general solution. (a) \(\frac{d y}{d t}=-y\) (b) \(\frac{d y}{d t}=-t\) (c) \(\frac{d y}{d t}=e^{-t}\)

Each function below is a solution to one of the second orderdifferential equations listed. To each function match the appropriate differential equation. \(C_{1}\) and \(C_{2}\) are constants. Differential Equations I. \(\frac{d^{2} x}{d t^{2}}-9 x=0\) II. \(\frac{d^{2} x}{d t^{2}}+9 x=0\) III. \(\frac{d^{2} x}{d t^{2}}=3 x\) Solution Functions (a) \(x(t)=5 e^{3 t}\) (b) \(x(t)=-2 e^{\sqrt{3} t}\) (c) \(x(t)=7 \sin 3 t\) (d) \(x(t)=C_{1} \sin 3 t+C_{2} \cos 3 t\) (e) \(x(t)=C_{1} e^{\sqrt{3} t}+C_{2} e^{-\sqrt{3} t}\)

For what value(s) of \(\beta\), if any, is (a) \(y=C_{1} \sin \beta t\) a solution to \(y^{\prime \prime}=16 y\) ? (b) \(y=C_{2} \cos \beta t\) a solution to \(y^{\prime \prime}=16 y\) ? (c) \(y=C_{3} e^{\beta t}\) a solution to \(y^{\prime \prime}=16 y\) ?

For what value(s) of \(\beta\), if any, is (a) \(y=C_{1} \sin \beta t\) a solution to \(y^{\prime \prime}=-16 y ?\) (b) \(y=C_{2} \cos \beta t\) a solution to \(y^{\prime \prime}=-16 y\) ? (c) \(y=C_{3} e^{\beta t}\) a solution to \(y^{\prime \prime}=-16 y\) ?

(a) There are two values of \(\lambda\) such that \(y=e^{\lambda^{t}}\) is a solution to \(y^{\prime \prime}+7 y^{\prime}+12 y=0\). Find them and label them \(\lambda\), and \(\lambda_{2}\). (b) Let \(y=C_{1} e^{\lambda_{1} I}+C_{2} e^{\lambda_{2} t}\), where \(C_{1}\) and \(C_{2}\) are arbitrary constants. Verify that \(y(t)\) is a solution to \(y^{\prime \prime}+7 y^{\prime}+12 y=0\).

(a) Find \(\lambda\) such that \(y=e^{\lambda t}\) is a solution to \(y^{\prime \prime}+4 y^{\prime}+4 y=0\). (b) Verify that \(y=t e^{\lambda t}\) is also a solution to \(y^{\prime \prime}+4 y^{\prime}+4 y+0\).