Calculus: Early Transcendentals

James Stewart

Chapter 9

Differential Equations - all with Video Answers

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Section 2

Direction Fields and Euler's Method

Problem 1

A direction field for the differential equation $y^{\prime}=x \cos \pi y$ is shown.
(a) Sketch the graphs of the solutions that satisfy the given initial conditions.
(i) $y(0)=0$
(ii) $y(0)=0.5$
(iii) $y(0)=1$
(iv) $y(0)=1.6$
(b) Find all the equilibrium solutions.

Chris Trentman

Chris Trentman

Numerade Educator

Problem 1

A direction field for the differential equation $y^{\prime}=x \cos \pi y$ is shown.
(a) Sketch the graphs of the solutions that satisfy the given initial conditions.
(i) $y(0)=0$
(ii) $y(0)=0.5$
(iii) $y(0)=1$
(iv) $y(0)=1.6$
(b) Find all the equilibrium solutions.

Chris Trentman

Numerade Educator

Problem 2

A direction field for the differential equation $ y^1 = x \sin y $ is shown.
(a) Sketch the graphs of the solutions that satisfy the given initial conditions.
(i) $y(0)=1$
(ii) $y(0)=0.2$
(iii) $y(0)=2$
(iv) $y(1)=3$
(b) Find all the equilibrium solutions.

Chris Trentman

Numerade Educator

Problem 3

Match the differential equation with its direction field (labeled I-IV). Give reasons for your answer.
$ y' = 2 - y $

Chris Trentman

Numerade Educator

Problem 4

Match the differential equation with its direction field (labeled I-IV). Give reasons for your answer.
$ y' = x(2 - y) $

Chris Trentman

Numerade Educator

Problem 5

Match the differential equation with its direction field (labeled I-IV). Give reasons for your answer.
$ y' = x + y - 1 $

Chris Trentman

Numerade Educator

Problem 6

Match the differential equation with its direction field (labeled I-IV). Give reasons for your answer.
$ y' = \sin x \sin y $

Carson Merrill

Numerade Educator

Problem 7

Use the direction field labeled I (above) to sketch the graphs of the solutions that satisfy the given initial conditions.
(a) $y(0)=1$
(b) $y(0)=2.5$
(c) $y(0)=3.5$

Chris Trentman

Numerade Educator

Problem 8

Use the direction field labeled III (above) to sketch the graphs of the solutions that satisfy the given initial conditions.
(a) $y(0)=1$
(b) $y(0)=2.5$
(c) $y(0)-3.5$

Chris Trentman

Numerade Educator

Problem 9

Sketch a direction field for the differential equation. Then use it to sketch three solution curves.
$ y' = 1 + y $

Chris Trentman

Numerade Educator

Problem 10

Sketch a direction field for the differential equation. Then use it to sketch three solution curves.
$$
y^{\prime}=x-y+1
$$

Chris Trentman

Numerade Educator

Problem 11

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.
$ y' = y - 2x, (1,0) $

Chris Trentman

Numerade Educator

Problem 12

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.
$ y' = 1 - xy, (0,0) $

Chris Trentman

Numerade Educator

Problem 13

Sketch the direction field of the differential equation. Then use it to sketch a solution curve that passes through the given point.
$ y' = y + xy, (0,0) $

Chris Trentman

Numerade Educator

Problem 14

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

Chris Trentman

Numerade Educator

Problem 15

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 $

Robert Barker

Numerade Educator

Problem 16

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) $

Kirsten Bauck

Numerade Educator

Problem 17

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?

Mike Gaerlan

Numerade Educator

Problem 18

Make a rough sketch of a direction field for the autonomous differential equation $ y' = f(y), $ where the graph of $ y $ is as shown. How does the limiting behavior of solutions depend on the value of $ y(0)? $

Chris Trentman

Numerade Educator

Problem 19

(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?

Chris Trentman

Numerade Educator

Problem 20

A direction field for a differential equation is shown. Draw, with a ruler, the graphs of the Euler approximations to the solution curve that passes through the origin. Use step sizes $ h = 1 $ and $ h = 0.5. $ Will the Euler estimates be under-estimates or overestimates? Explain.

Madi Sousa

Numerade Educator

Problem 21

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. $

Chris Trentman

Numerade Educator

Problem 22

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

Chris Trentman

Numerade Educator

Problem 23

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. $

Chris Trentman

Numerade Educator

Problem 24

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.

Chris Trentman

Numerade Educator

Problem 25

(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?

EI

Eric Icaza

Numerade Educator

Problem 26

(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.

EI

Eric Icaza

Numerade Educator

Problem 27

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.

Chris Trentman

Numerade Educator

Problem 28

In Exercise 14 in Section 9.1 we considered a $ 95^oC $ cup of coffee in a $ 20^oC $ room. Suppose it is known that the coffee cools at a rate of $ 1^oC $ per minute when its temperature is $ 70^oC. $
(a) What does the differential equation become in this case?
(b) Sketch a direction field and use it to sketch the solution curve for the initial-value problem. What is the limiting value of the temperature?
(c) Use Euler's method with step size $ h = 2 $ minutes to estimate the temperature of the coffee after 10 minutes.

Kirsten Bauck

Numerade Educator

Problem 28

In Exercise 14 in Section 9.1 we considered a $ 95^oC $ cup of coffee in a $ 20^oC $ room. Suppose it is known that the coffee cools at a rate of $ 1^oC $ per minute when its temperature is $ 70^oC. $
(a) What does the differential equation become in this case?
(b) Sketch a direction field and use it to sketch the solution curve for the initial-value problem. What is the limiting value of the temperature?
(c) Use Euler's method with step size $ h = 2 $ minutes to estimate the temperature of the coffee after 10 minutes.

Kirsten Bauck

Numerade Educator