Chapter 9

Solve the differential equation. \( t \ln t \frac {dr}{dt} + r = te^t \)

Find the solution of the differential equation that satisfies the given initial condition. \( x + 3y^2 \sqrt {x^2 + 1} \frac {dy}{dx} = 0, y(0) = 1 \)

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point. $$ y^{\prime}=x+y^{2}, \quad(0,0) $$

Suppose you have just poured a cup of freshly brewed coffee with temperature \( 95^{\circ} \) in a room where the temperature is \( 20^{\circ}. \) (a) When do you think the coffee cools most quickly? What happens to the rate of cooling as time goes by? Explain. (b) Newtons Law of Cooling states that the rate of cooling of an object is proportional to the temperature difference between the object and its surroundings, provided that this difference is not too large. Write a differential equation that expresses Newtons Law of Cooling for this particular situation. What is the initial condition? In view of your answer to part (a), do you think this differential equation is an appropriate model for cooling? (c) Make a rough sketch of the graph of the solution of the initial-value problem in part (b).

Solve the initial-value problem. \( x^2y' + 2xy = \ln x, y(1) = 2 \)

Find the solution of the differential equation that satisfies the given initial condition. \( x \ln x = y(1 + \sqrt {3 + y^2}) y', y(1) = 1 \)

Use a computer algebra system to draw a direction field for the given differential equation. Ger a printout and sketch on it the solution curve that passes through (0,1). Then use the CAS to draw the solution curve and compare it with your sketch. \( y' = x^2 \sin y \)

The table gives the midyear population of Japan, in thousands, from 1960 to 2010. Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both function, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose \( t = 0 \) to correspond to 1960 or 1980.]

Find the solution of the differential equation that satisfies the given initial condition. \( \frac {dP}{dt} = \sqrt {Pt}, P(1) = 2 \)

Use a computer algebra system to draw a direction field for the given differential equation. Ger a printout and sketch on it the solution curve that passes through (0,1). Then use the CAS to draw the solution curve and compare it with your sketch. \( y' = x(y^2 - 4) \)

Von Bertalanffys equation states that the rate of growth in length of an individual fish is proportional to the difference between the current lenght \( L \) and the asymptotic length \( L_x \) (in centimeters). (a) Write a differential equation that expresses this idea. (b) Make a rough sketch of the graph of a solution of a typical initial-value problem for this differential equation.

Solve the initial-value problem. \( t \frac {du}{dt} = t^2 + 3u, t > 0, u(2) = 4 \)

Consider a population \( P = P(t) \) with constant relative birth and death rates \( \alpha , \beta, \) respectively, and a constant emigration rate \( m, \) where \( \alpha, \beta, \) and \( m \) are positive constants. Assume that \( \alpha > \beta. \) Then the rate of change of the population at time \( t \) is modeled by the differential equation \( \frac {dP}{dt} = kP - m \) where \( k = \alpha - \beta \) (a) Find the solution if this equation that satisfies the initial condition \( P(0) = P_o. \) (b) What condition on \( m \) will lead to an exponential expansion of the population? (c) What condition on \( m \) will result in a constant population? A population decline? (d) In 1847, the population of Ireland was about 8 million and the difference between the relative birth and death rates was \( 1.6% \) of the population. Because of potato famine in the 1840s and 1850s, about 210,000 inhabitants per year emigrated from Ireland. Was the population expanding or declining at that time?

Find the solution of the differential equation that satisfies the given initial condition. \( y' \tan x = a + y, y(\pi/3) = a, 0 < x < \pi/2 \)

Use a computer algebra system to draw a direction field for the differential equation \( y' = y^3 - 4y. \) Get a printout and sketch on it solutions that satisfy the initial condition \( y(0) = c \) for various of \( c. \) For what values of \( c \) does \( \lim \) \(_{t \to\infty} y(t) \) exist? What are the possible values for this limit?

Differential equations have been used extensively in the study of drug dissolution for patients given oral medication. One such equation is the Weibull equation for the concentration c(t) of the drug: \( \frac {dc}{dt} = \frac {k}{t^b} (c_a - c) \) where \( k \) and \( c_a \) are positive constants and \( 0 < b < 1\. \) Verify that \( c(t) = c_a (1 - e^{-at^{1-b}}) \) is a solution of the Weibull equation for \( 1 > 0, \) where \( a = k/(1 - b). \) What does the differential equation say about how drug dissolution occurs?

Solve the initial-value problem. \( xy' + y = x \ln x, y(1) = 0 \)

Let \( c \) be a positive number. A differential equation of the form \( \frac {dy}{dt} = ky^{1+c} \) where \( k \) is a positive constant, is called a doomsday equation because the exponent in the expression \( ky^{1+c} \) is larger than the exponent 1 for natural growth. (a) Determine the solution that satisfies the initial condition \( y(0) = y_0. \) (b) Show that there is a finite time \( t = T \) (doomsday) such that lim \( _{t \to T} - y(t) = \infty. \) (c) An especially prolific breed of rabbits has the growth term \( ky^{1.01}. \) If 2 such rabbits breed initially and the warren has 16 rabbits after three months, then when is doomsday?

Find the solution of the differential equation that satisfies the given initial condition. \( \frac {dL}{dt} = kL^2\ln t, L(1) = -1 \)

Solve the initial-value problem. \( xy' = y + x^2 \sin x, y(\pi) = 0 \)

Let's modify the logistic differential equation of Example 1 as follows: \( \frac {dP}{dt} = 0.08P ( 1 - \frac {P}{1000}) - 15 \) (a) Suppose \( P(t) \) represents a fish population at time \( t, \) where \( t \) is measured in weeks. Explain the meaning of the final term in the equation (-15). (b) Draw a direction field for this differential equation. (c) What are the equilibrium solutions? (d) Use the direction field to sketch several solution curves. Describe what happens to the fish population for various initial populations. (e) Solve this differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial populations 200 and 300. Graph the solutions and compare with your sketches in part (d).

Find an equation of the curve that passes through the point \( (0,2) \) and whose slope at \( (x, y) \) is \( x/y. \)

(a) Use Euler's method with each of the following step sizes to estimate the value of \( y(0.4), \) where \( y \) is the solution of the initial-value problem \( y' = y, y(0) = 1. \) (i) \( h = 0.4 \) (ii) \( h = 0.2 \) (iii) \( h = 0.1 \) (b) We know that the exact solution of the initial-value problem in part (a) is \( y = e^x. \) Draw, as accurately as you can, the graph of \( y = e^x, 0 \le x \le 0.4, \) together with the Euler approximations using the step sizes in part (a). (Your sketches should resemble Figure 12, 13, and 14.) Use your sketches to decide whether your estimates in part (a) are underestimates or overestimates. (c) The error in Euler's method is the difference between the exact value and the approximate value. Find the errors made in part (a) in using Euler's method to estimate the true value of \( y(0.4), \) namely \( e^{0.4}. \) What happens to the errors each time the steps size is halved?

Solve the initial-value problem. \( (x^2 + 1) \frac {dy}{dx} + 3x(y -1) = 0, y(0) = 2 \)

Find the function \( f \) such that \( f'(x) = xf(x) - x \) and \( f(0) = 2. \)

Solve the differential equation and use a calculator to graph several members of the family of solutions. How does the solution curve change as \( C \) varies? \( xy' + 2y = e^x \)

There is considerable evidence to support the theory that for some species there is a minimum population \( m \) such that the species will become extinct if the size of the population falls below \( m. \) This condition can be incorporated into the logistic equation by introducing the factor \( (1 - m/P). \) Thus the modified logistic model is given by the differential equation \( \frac {dP}{dt} = kP (1 - \frac {P}{M})(1 - \frac {m}{P}) \) (a) Use the differential equation to show that any solution is increasing if \( m < P < M \) and decreasing if \( 0 < P < m. \) (b) For the case where \( k = 0.08, M = 1000, \) and \( m = 200, \) draw a direction field and use it to sketch several solution curves. Describe what happens to the population for various initial populations. What are the equilibrium solutions? (c) Solve the differential equation explicitly, either by using partial fractions or with a computer algebra system. Use the initial population \( P_0. \) (d) Use the solution in part (c) to show that if \( P_0 < m, \) then the species will become extinct. [Hint: Show that the numerator in your expression for \( P(t) \) is 0 for some value of \( t. \) ]

Solve the differential equation \( y' = x + y \) by making the change of variable \( u = x + y. \)

Use Euler's method with step size 0.5 to compute the approximate y-values \( y_1, y_2, y_3, \) and \( y_4 \) of the solution of the initial-value problem \( y' = y - 2x, y(1) = 0. \)

Solve the differential equation and use a calculator to graph several members of the family of solutions. How does the solution curve change as \( C \) varies? \( xy' = x^2 + 2y \)

Solve the differential equation \( xy' = y + xe^{y/x} \) by making the change of variable \( v = y/x. \)

Use Euler's method with step size 0.2 to estimate \(y(1)\) where \(y(x)\) is the solution of the initial-value problem \(y^{\prime}=x^{2} y-\frac{1}{2} y^{2}, y(0)=1\)

A Bernoulli differential equation (named after James Bernoulli) is of the form \( \frac {dy}{dx} + P(x)y = Q(x)y^n \) Observe that, if \( n = 0 \) or 1, the Bernoulli equation is linear. For other values of \( n, \) show that the substitution \( u = y^{1-n} \) transforms the Bernoulli equation into the linear equation \( \frac {du}{dx} + ( 1 - n)P(x)u = (1 - n)Q(x) \)

In a seasonal-growth model, a period function of time is introduced to account for seasonal variations in the rate of growth. Such variations could, for examples, be caused by seasonal changes in the availability of food. (a) Find the solution of the seasonal-growth model \( \frac {dP}{dt} = kP \cos(rt - \phi) P(0) = P_0 \) where \( k, r, \) and \( \phi \) are positive constants. (b) By graphing the solution for several values of \( k, r, \) and \( \phi, \) explain how to the values of \( k, r, \) and \( \phi \) affect the solution. What can you say about lim \( _{t \to \infty} P(t)? \)

(a) Solve the differential equation \( y' = 2x \sqrt {1 - y^2}. \) (b) Solve the initial-value problem \( y' = 2x \sqrt {1 - y^2}, y(0) = 0, \) and graph the solution. (c) Does the initial-value problem \( y' = 2x \sqrt {1 - y^2}, y(0) = 2, \) have a solution? Explain.

Use Euler's method with step size 0.1 to estimate \( y(0.5), \) where \( y(x) \) is the solution of the initial-value problem \( y' = y + xy, y(0) = 1. \)

a) Use Euler's method with step size 0.2 to estimate \( y(1.4), \) where \( y(x) \) is the solution of the initial-value problem \( y' = x - xy, y(1) = 0. \) (b) Repeat part (a) with step size 0.1.

Solve the equation \( e^{-y}y' + \cos x = 0 \) and graph several members of the family of solutions. How does the solution curve change as the \( C \) varies?

Solve the initial-value problem \( y' = (\sin x) \sin y, y(0) = \pi/2, \) and graph the solution (if your CAS does implicit plots).

(a) Program a calculator or computer to use Euler's method to compute \( y(1), \) where \( y(x) \) is the solution of the initial-value problem \( \frac {dy}{dx} + 3x^2y = 6x^2 y(0) = 3 \) (i) \( h = 1 \) (ii) \( h = 0.1 \) (iii) \( h = 0.01 \) (iv) \( h = 0.001 \) (b) Verify that \( y = 2 + e^{-x^3} \) is the exact solution of the differential equation. (c) Find the errors in using Euler's method to compute \( y(1) \) with the step sizes in part (a). What happens to the error when the step size is divided by 10?

Solve the second-order equation \( xy'' + 2y' = 12x^2 \) by making the substitution \( u = y'. \)

(a) Program your computer algebra system, using Euler's method with step 0.01, to calculate \( y(2), \) where \( y \) is the solution of the initial-value problem \( y' = x^3 - y^3 y(0) = 1 \) (b) Check your work by using the CAS to draw the solution curve.

The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of \( C \) farads \( (F), \) and a resistor with a resistance of \( R \) ohms \( (\Omega). \) The voltage drop across the capacitor is \( Q/C, \) where \( Q \) is the charge (in coulombs), so in this case Kirchhoff's Law gives \( RI + \frac {Q}{C} = E(t) \) But \( I = dQ/dt, \) so we have \( R \frac {dQ}{dt} + \frac {1}{C} Q = E(t) \) Suppose the resistance is 5 \( \Omega, \) the capacitance is 0.05 F, and a battery gives a constant voltage of 60V. (a) Draw a direction field for this differential equation. (b) What is the limiting value of the charge? (c) Is there an equilibrium solution? (d) If the initial charge is \( Q(0) = 0 C, \) use the direction field to sketch the solution curve. (e) If the initial charge is \( Q(0) = 0 C, \) use Euler's method with step size 0.1 to estimate the charge after half a second.

(a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b) Solve the differential equation. (c) Use the CAS to draw several members of the family of solutions obtained in part (b). Compare with the curves from part (a). \( y' = y^2 \)

(a) Use a computer algebra system to draw a direction field for the differential equation. Get a printout and use it to sketch some solution curves without solving the differential equation. (b) Solve the differential equation. (c) Use the CAS to draw several members of the family of solutions obtained in part (b). Compare with the curves from part (a). \( y' = xy \)

The figure shows a circuit containing an electromotive force, a capacitor with a capacitance of \( C \) farads \( (F), \) and a resistor with resistance of \( R \) ohms \( (\Omega). \) The voltage drop across is \( Q/C, \) where \( Q \) is the charge (in coulombs), so in this case Kirchhoff's Law gives \( RI + \frac {Q}{C} = E(t) \) But \( I = dQ/dt \) (see Examples 3.7.3), so we have \( R \frac {dQ}{dt} + \frac {1}{C}Q = E(t) \) Suppose the resistance is \( 5 \Omega, \) the capacitance is 0.05 F, a battery gives a constant voltage of 60 V, and the initial charge is \( Q(0) = 0 C. \) Find the charge and the current at time \( t. \)

Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. \( x^2 + 2y^2 = k^2 \)

Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. \( y^2 = kx^3 \)

Let \( P(t) \) be the performance level of someone learning a skill as a function of the training time \( t. \) The graph of \( P \) is called a learning curve. In Exercise 9.1.15 we proposed the differential equation \( \frac {dP}{dt} = k[M - P(t)] \) as a reasonable model for learning, where \( k \) is a positive constant. Solve it as a linear differential equation and use your solution to graph the learning curve.

Find the orthogonal trajectories of the family of curves. Use a graphing device to draw several members of each family on a common screen. \( y = \frac {k}{x} \)