Homework Homework Assignment #19 Read Section 9.3 Page 521, Exercises: 1 – 41(EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and

Homework

Homework Assignment #19 Read Section 9.3 Page 521, Exercises: 1 – 41(EOO) Quiz next time

Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 521Which of the following differential equations are first order?


2

2

3

2

1. a

b

c sin

d x sin

e 3

f y 0

x

y x

y y

y yy x

y e y x

yy y

xy x y

Selections a, c, d, and f are first order differentialequations.

Homework, Page 521Verify that each given function is a solution of the differential equation.

5.


24 0, 12 4yy x y x

2

2

2

2

2

1 812 4

2 12 4

1 84 12 4 4 4 4 0

2 12 4

12 4 is a solution of 4 0

xy x y

x

xyy x x x x x

x

y x yy x

Homework, Page 521Verify that the given function is a solution of the differential equation.

9. 2 5 0, sin 2xy y y y e x

sin 2 2 cos 2 sin 2

4 sin 2 2 cos 2 2 cos 2 sin 2

3 sin 2 4 cos

2 5 3 sin 2 4 cos 2 2 cos 2 sin 2

5 sin 0

sin is a solution of 2

x x x

x x x x

x x

x x x x

x

x

y e x y e x e x

y e x e x e x e x

e x e x

y y y e x e x e x e x

e x

y e x y

5 0y y Rogawski CalculusCopyright © 2008 W. H. Freeman and Company

Homework, Page 521Solve, using separation of variables.

13. 2y xy

2 22

22

2

1 1

2

2

dy dyy xy xy xdx

dx y

dyxdx x C

yy

yx C



17. 2 6 4 0dy

ydx

6 42 6 4 0

2 3 2

2ln 3 2 3 2

3

xx

dy dy y dyy dx

dx dx y

Cey x C y Ce y



21. 2 21y y x

3

2 2 22

3

3

11 1

3

1 3 3

3 3 3

dy xy y x x dx x C

yy

x xC y

y x x C



25. secy x y

2

1 2

sec cossec

1cos sin

2

1sin

2

dy dyx y xdx ydy xdx

dx y

ydy xdx y x C

y x C


Homework, Page 521Solve the initial value problem.

29. 2 0, ln 2 3y y y

2

2 ln 22

ln 2 2

2 0 2 2 2

ln 2 3

13 3 12 12

4

x

x

dy dy dy dyy y dx dx

dx dx y y

y x C y Ce y Ce

Ce C C y e



33. 1 2 , 0 3y x y y

22 2

2

2

1 1 1 0 02 2 2

12

1 12 2

1ln 2

2

2 2 2 3

1 2

x x x x

x x

dy dyx dx x dx

y y

y x x C

y Ce y Ce Ce

C y e



37. 2 1 , 1 0dy

t t y ty ydt

2 2

2 2

2 2

1 1

1 1 1 1

1 1

1 1

1ln 1 ln

dy dyt y ty t t y t ty

dt dtdy dy

t y t y t t ydt dt

dy t dy tdt dt

y yt t

y t Ct


Homework, Page 52137. Continued.

1 1ln ln

ln1 ln111

1

1ln 1 ln

1 1

1 0 1

1

t C tt t

t

t

t

y t Ct

y e y Ce

eeCe C e y

e

ety

e



41. 2 sin , 0 3y y x y

2 2

1sin sin cos

1 1 13 3

cos cos 0 1

2 13 3 1 3 2

23 cos 3

dy dyxdx xdx x C

yy y

yx C C C

C C C yx



Jon Rogawski

Calculus, ETFirst Edition

Chapter 9: Introduction to Differential Equations

Section 9.3: Graphical and Numerical Models


Most differential equations cannot be solved explicitly, but there aregraphical and numerical methods that afford estimates sufficientlyaccurate for most requirements.

In this section, the derivative relative to time will be denoted using the

notation from some physics and engineering disciplines or dy

ydt

The function , is a "set of instructions" which defines

the graph of the solution to the differential equation. This graph is called

a slope field and it consists of short line segments thr

dyt F ty

dt

ough evenly spaced

points defined on the x-y plane.



Definition – Isocline

An isocline is the set of points on the x-y coordinate plane where the slope defined by the differential equation has a constant value.

The next slide illustrates the process of drawing a slope field using isoclines.




7. Sketch the slope field of for 2 , 2. Based on the sketch,

determine lim , where is a solution with 0 0. What is

lim if 0 0?t

t

y ty t y

y t y t y

y t y

x

y


The above slope field shows the rate of warming or cooling for an object placed intoan environment at 40°F.


Most differential equations have a uniqueness property, that is, there is a unique solution to the differential equation meeting a specific initial value criterion. The graphbelow is that of an exception to this property.


Given a differential equation and an initial condition, we may use Euler’s method to approximate the function’s value at a nearby value of t using the formula:

,y F t y




Using a Computer Algebra System or CAS, it is easy to rapidly evaluate Euler’s Method results for large numbers of very small increments. As shown on an earlierslide, the more numerous and smaller the intervals, the more accurate the result. The table below shows the results for a CAS evaluation of an Euler’s solution.

Use Euler’s Method with h = 0.1 to approximate the given value of y(t).


214. 0.5 ; , 0 0y y y y

Homework

Homework Assignment #20 Review Section 9.3 Page 537, Exercises: 1 – 17 (EOO) Quiz next time


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Homework Homework Assignment #19 Read Section 9.3 Page 521, Exercises: 1 – 41(EOO) Quiz next time Rogawski Calculus Copyright © 2008 W. H. Freeman and