Chapter 11

(a) Show that \(P=1 /\left(1+e^{-t}\right)\) satisfies the logistic equation $$\frac{d P}{d t}=P(1-P)$$ (b) What is the limiting value of \(P\) as \(t \rightarrow \infty ?\)

Determine which of the following differential equations are separable. Do not solve the equations. (a) \(y^{\prime}=y\) (b) \(y^{\prime}=x+y\) (c) \(y^{\prime}=x y\) (d) \(y^{\prime}=\sin (x+y)\) (e) \(y^{\prime}-x y=0\) (f) \( y^{\prime}=y / x\) (g) \(y^{\prime}=\ln (x y)\) (h) \(y^{\prime}=(\sin x)(\cos y)\) (i) \(y^{\prime}=(\sin x)(\cos x y)\) (j) \( y^{\prime}=x / y\) (k) \( y^{\prime}=2 x\) (l) \(y^{\prime}=(x+y) /(x+2 y)\)

Using Euler's method, complete the following table for \(y^{\prime}=(x-2)(y-3)\) $$\begin{array}{c|c|c} \hline x & y & y^{\prime} \\ \hline 0.0 & 4.0 & \\ \hline 0.1 & & \\ \hline 0.2 & & \\ \hline \end{array}$$

(a) For \(d y / d x=x^{2}-y^{2},\) find the slope at the following points: \((1,0), \quad(0,1), \quad(1,1), \quad(2,1), \quad(1,2), \quad(2,2)\) (b) Sketch the slope field at these points.

Is \(y=x^{3}\) a solution to the differential equation $$x y^{\prime}-3 y=0 ?$$

A quantity \(P\) satisfies the differential equation \(\frac{d P}{d t}=k P\left(1-\frac{P}{100}\right)\) Sketch approximate solutions satisfying each of the following initial conditions: (a) \(\quad P_{0}=8\) (b) \(\quad P_{0}=70\) (c) \(\quad P_{0}=125\)

Write a differential equation for the balance \(B\) in an investment fund with time, \(t,\) measured in years. The balance is earning interest at a continuous rate of \(5 \%\) per year, and payments are being made out of the fund at a continuous rate of 12,000 dollars per year.

In Exercises \(2-28,\) use separation of variables to find the solutions to the differential equations subject to the given initial conditions. $$\frac{d P}{d t}=-2 P, \quad P(0)=1$$

Using Euler's method, complete the following table for \(y^{\prime}=4 x y\) $$\begin{array}{c|c|c} \hline x & y & y^{\prime} \\ \hline 1.00 & -3.0 & \\ \hline 1.01 & & \\ \hline 1.02 & & \\ \hline \end{array}$$

Determine whether each function is a solution to the differential equation and justify your answer: $$x \frac{d y}{d x}=4 y.$$ (a) \(y=x^{4}\) (b) \(y=x^{4}+3\) (c) \(y=x^{3}\) (d) \(y=7 x^{4}\)

A quantity \(Q\) satisfies the differential equation $$\frac{d Q}{d t}=k Q(1-0.0004 Q)$$ Sketch approximate solutions satisfying each of the following initial conditions: (a) \(Q_{0}=300\) (b) \(Q_{0}=1500\) (c) \(Q_{0}=3500\)

Write a differential equation for the balance \(B\) in an investment fund with time, \(t,\) measured in years. The balance is earning interest at a continuous rate of \(3.7 \%\) per year, and money is being added to the fund at a continuous rate of 5000 dollars per year.

In Exercises \(2-28,\) use separation of variables to find the solutions to the differential equations subject to the given initial conditions. $$\frac{d P}{d t}=0.02 P, \quad P(0)=20$$

A population, \(P\), in millions, is 1500 at time \(t=0\) and its growth is governed by $$\frac{d P}{d t}=0.00008 P(1900-P)$$ Use Euler's method with \(\Delta t=1\) to estimate \(P\) at time \(t=1,2,3\)

Determine whether each function is a solution to the differential equation and justify your answer:$$y \frac{d y}{d x}=6 x^{2}$$ (a) \(y=4 x^{3}\) (b) \(y=2 x^{3 / 2}\) (c) \(y=6 x^{3 / 2}\)

A quantity \(P\) satisfies the differential equation $$\frac{d P}{d t}=k P\left(1-\frac{P}{250}\right), \quad \text { with } k>0$$ Sketch a graph of \(d P / d t\) as a function of \(P\)

Write a differential equation for the balance \(B\) in an investment fund with time, \(t,\) measured in years. The balance is losing value at a continuous rate of \(8 \%\) per year, and money is being added to the fund at a continuous rate of 2000 dollars per year.

In Exercises \(2-28,\) use separation of variables to find the solutions to the differential equations subject to the given initial conditions. $$\frac{d L}{d p}=\frac{L}{2}, \quad L(0)=100$$

(a) Use Euler's method to approximate the value of \(y\) at \(x=1\) on the solution curve to the differential equation \(d y / d x=3\) that passes through \((0,2) .\) Use \(\Delta x=0.2\) (b) What is the solution to the differential equation \(d y / d x=3\) with initial condition \(y=2\) when \(x=0 ?\) (c) What is the error for the Euler's method approximation at \(x=1 ?\) (d) Explain why Euler's method is exact in this case.

Show that \(y(x)=A e^{\lambda x}\) is a solution to the equation \(y^{\prime}=\lambda y\) for any value of \(A.\)

Sketch solution curves with a variety of initial values for the differential equations. You do not need to find an equation for the solution. \(\frac{d y}{d t}=\alpha-y,\) where \(\alpha\) is a positive constant.

A quantity \(A\) satisfies the differential equation $$\frac{d A}{d t}=k A(1-0.0002 A), \quad \text { with } k>0$$ Sketch a graph of \(d A / d t\) as a function of \(A\)

Find all equilibrium points. Give answers as ordered pairs \((x, y).\) $$\begin{aligned} &\frac{d x}{d t}=-3 x+x y\\\ &\frac{d y}{d t}=5 y-x y \end{aligned}$$

Write a differential equation for the balance \(B\) in an investment fund with time, \(t,\) measured in years. The balance is losing value at a continuous rate of \(6.5 \%\) per year, and payments are being made out of the fund at a continuous rate of 50,000 dollars per year.

In Exercises \(2-28,\) use separation of variables to find the solutions to the differential equations subject to the given initial conditions. $$\frac{d Q}{d t}=\frac{Q}{5}, \quad Q=50 \text { when } t=0$$

(a) Use five steps of Euler's method to determine an approximate solution for the differential equation \(d y / d x=y-x\) with initial condition \(y(0)=10,\) using step size \(\Delta x=0.2 .\) What is the estimated value of \(y\) at \(x=1 ?\) (b) Does the solution to the differential equation appear to be concave up or concave down? (c) Are the approximate values overestimates or underestimates?

Show that \(y=\sin 2 t\) satisfies $$\frac{d^{2} y}{d t^{2}}+4 y=0.$$

Sketch solution curves with a variety of initial values for the differential equations. You do not need to find an equation for the solution. \(\frac{d w}{d t}=(w-3)(w-7)\)

Find all equilibrium points. Give answers as ordered pairs \((x, y).\) $$\begin{aligned} &\frac{d x}{d t}=-2 x+4 x y\\\ &\frac{d y}{d t}=-8 y+2 x y \end{aligned}$$

A bank account that earns \(10 \%\) interest compounded continuously has an initial balance of zero. Money is deposited into the account at a constant rate of 1000 dollars per year. (a) Write a differential equation that describes the rate of change of the balance \(B=f(t)\). (b) Solve the differential equation to find the balance as a function of time.

In Exercises \(2-28,\) use separation of variables to find the solutions to the differential equations subject to the given initial conditions. $$P \frac{d P}{d t}=1, P(0)=1$$

(a) Use ten steps of Euler's method to determine an approximate solution for the differential equation \(y^{\prime}=x^{3}, y(0)=0,\) using a step size \(\Delta x=0.1\) (b) What is the exact solution? Compare it to the computed approximation. (c) Use a sketch of the slope field for this equation to explain the results of part (b).

Show that, for any constant \(P_{0},\) the function \(P=P_{0} e^{t}\) satisfies the equation $$\frac{d P}{d t}=P.$$

Find all equilibrium points. Give answers as ordered pairs \((x, y).\) $$\begin{aligned} &\frac{d x}{d t}=15 x-5 x y\\\ &\frac{d y}{d t}=10 y+2 x y \end{aligned}$$

At time \(t=0,\) a bottle of juice at \(90^{\circ} \mathrm{F}\) is stood in a mountain stream whose temperature is \(50^{\circ} \mathrm{F}\). After 5 minutes, its temperature is \(80^{\circ} \mathrm{F}\). Let \(H(t)\) denote the temperature of the juice at time \(t,\) in minutes. (a) Write a differential equation for \(H(t)\) using Newton's Law of Cooling. (b) Solve the differential equation. (c) When will the temperature of the juice have dropped to \(60^{\circ} \mathrm{F} ?\)

In Exercises \(2-28,\) use separation of variables to find the solutions to the differential equations subject to the given initial conditions. $$\frac{d m}{d t}=3 m, \quad m=5 \text { when } t=1$$

Use implicit differentiation to show that \(x^{2}+y^{2}=r^{2}\) is a solution to the differential equation \(d y / d x=-x / y.\)

(a) Find the equilibrium solution to the differential equation $$\frac{d y}{d t}=0.5 y-250$$ (b) Find the general solution to this differential equation. (c) Sketch the graphs of several solutions to this differential equation, using different initial values for \(y.\) (d) Is the equilibrium solution stable or unstable?

Find all equilibrium points. Give answers as ordered pairs \((x, y).\) $$\begin{aligned} &\frac{d x}{d t}=x^{2}-x y\\\ &\frac{d y}{d t}=15 y-3 y^{2} \end{aligned}$$

The velocity, \(v,\) of a dust particle of mass \(m\) and acceleration \(a\) satisfies the equation $$m a=m \frac{d v}{d t}=m g-k v, \quad \text{ where \(g,k\) are constant. }$$ By differentiating this equation, find a differential equation satisfied by \(a\). (Your answer may contain \(m, g, k\) but not \(v\).) Solve for \(a\), given that \(a(0)=g\).

In Exercises \(2-28,\) use separation of variables to find the solutions to the differential equations subject to the given initial conditions. $$\frac{d I}{d x}=0.2 I, \quad I=6 \text { where } x=-1$$

A quantity \(Q\) satisfies the differential equation $$\frac{d Q}{d t}=\frac{t}{Q}-0.5.$$ (a) If \(Q=8\) when \(t=2,\) use \(d Q / d t\) to determine whether \(Q\) is increasing or decreasing at \(t=2.\) (b) Use your work in part (a) to estimate the value of \(Q\) when \(t=3 .\) Assume the rate of change stays approximately constant over the interval from \(t=2\) to \(t=3.\)

(a) Find all equilibrium solutions for the differential equation $$\frac{d y}{d x}=0.5 y(y-4)(2+y)$$ (b) Draw a slope field and use it to determine whether each equilibrium solution is stable or unstable.

Given the system of differential equations $$\begin{aligned} &\frac{d x}{d t}=5 x-3 x y\\\ &\frac{d y}{d t}=-8 y+x y \end{aligned}$$ determine whether \(x\) and \(y\) are increasing or decreasing at the point (a) \(\quad x=3, y=2\) (b) \(x=5, y=1\)

A deposit is made to a bank account paying \(8 \%\) interest compounded continuously. Payments totaling 2000 dollars per year are made from this account. (a) Write a differential equation for the balance, \(B,\) in the account after \(t\) years. (b) Find the equilibrium solution of the differential equation. Is the equilibrium stable or unstable? Explain what happens to an account that begins with slightly more money or slightly less money than the equilibrium value. (c) Write the solution to the differential equation. (d) How much is in the account after 5 years if the initial deposit is (i) \(20,000 ?\) dollars (ii) \(30,000 ?\) dollars

Consider the solution of the differential equation \(y^{\prime}=y\) passing through \(y(0)=1\) (a) Sketch the slope field for this differential equation, and sketch the solution passing through the point (0,1) (b) Use Euler's method with step size \(\Delta x=0.1\) to estimate the solution at \(x=0.1,0.2, \ldots, 1\) (c) Plot the estimated solution on the slope field; compare the solution and the slope field. (d) Check that \(y=e^{x}\) is the solution of \(y^{\prime}=y\) with \(y(0)=1\)

In Exercises \(2-28,\) use separation of variables to find the solutions to the differential equations subject to the given initial conditions. $$\frac{1}{z} \frac{d z}{d t}=5, \quad z(1)=5$$

Fill in the missing values in the table given if you know that \(d y / d t=0.5 y .\) Assume the rate of growth given by dy/dt is approximately constant over each unit time interval and that the initial value of \(y\) is 8. $$\begin{array}{c|c|c|c|c|c}\hline t & 0 & 1 & 2 & 3 & 4 \\\\\hline y & 8 & & & & \\\\\hline \end{array}$$

(a) A cup of coffee is made with boiling water and stands in a room where the temperature is \(20^{\circ} \mathrm{C}\) If \(H(t)\) is the temperature of the coffee at time \(t,\) in minutes, explain what the differential equation $$\frac{d H}{d t}=-k(H-20)$$ says in everyday terms. What is the sign of \(k ?\) (b) Solve this differential equation. If the coffee cools to \(90^{\circ} \mathrm{C}\) in 2 minutes, how long will it take to cool to \(60^{\circ} \mathrm{C}\) degrees?

Given the system of differential equations$$\begin{array}{l} \frac{d P}{d t}=2 P-10 \\ \frac{d Q}{d t}=Q-0.2 P Q \end{array}$$ determine whether \(P\) and \(Q\) are increasing or decreasing at the point (a) \(\quad P=2, Q=3\) (b) \(\quad P=6, Q=5\)