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2x + y
dx 3 + 3 y2 - x
dy 2xy + y2 + 1
x2 + 2xy
7 di= x
dx x2 y + y3 dy 2xy + 1
2. (x + y) dx - (x - y) dy = 0
y(0) = 0 4. (x + ey) dy - dx = 0
6. x + xy = 1 - y, y(1) = 0
Hint: Let u = x2.
dy sin x
x-f + 2 y = ---------------,
y(2) = 1
x2 + 2 y
11. (x2 + y) dx + (x + ey) dy = 0
13. xdy- ydx = (xy)l/2 dx
14. (x + y) dx + (x + 2y) dy = 0,
15. ^ + 1) = y - yex
T = e2x + 3y y
10. (3y2 + 2xy) dx - (2xy + x2) dy = 0 dy 1
12 dx + y = T+ex
y(2) = 3
xdy - ydx = 2x2 y2 dy, y(1) = -2
16 dy _ x2 + y2 . dx x2 18. (2y + 3x) dx = xdy
20. / = e^
21. x/ = y + xey/x 23. x/ + y - y2e2x = 0
dy x2 - 1
y(-1) = 1
dx y2 + 1
24. 2 sin y cos x dx + cos y sin xdy = 0
25 (2y - xr+j)dx + (7+7 - 7jdy-0 26. (2y + 1) dx 7 X y) dy = 0
27. (cos 2y sinx) dx 2 tan xsin 2ydy = 0
28 3X2 2 y y3 29 dy = 2y + V x2 y2
' dx 2x + 3xy2 ' dx 2x
30. , y(0) = 1
dx 1 2xy2
31. (x2y + xy y) dx + (x2y 2x2) dy = 0
^ dy 3x2 y + y2 o
32. =------------5--------, y(1) = 2
dx 2x^ + 3xy
Two books mentioned in Section 2.3 are:
Bailey, N. T. J., The Mathematical Theory of Infectious Diseases and Its Applications (2nd ed.) (New York: Hafner Press, 1975).
Clark, Colin W., Mathematical Bioeconomics (2nd ed.) (New York: Wiley-Interscience, 1990).
An introduction to population dynamics in general is:
Frauenthal, J. C., Introduction to Population Modeling (Boston: Birkhauser, 1980).
A fuller discussion of the proof of the fundamental existence and uniqueness theorem can be found in many more advanced books on differential equations. Two that are reasonably accessible to elementary readers are:
Coddington, E. A., An Introduction to Ordinary Differential Equations (Englewood Cliffs, NJ: Prentice Hall, 1961; New York: Dover, 1989).
Brauer, F., and Nohel, J., The Qualitative Theory of Ordinary Differential Equations (New York: Benjamin, 1969; New York: Dover, 1989).
A valuable compendium of methods for solving differential equations is:
Zwillinger, D., Handbook of Differential Equations (3rd ed.) (San Diego: Academic Press, 1998).
A general reference on difference equations is:
Mickens, R. E., Difference Equations, Theory and Applications (2nd ed.) (New York: Van Nostrand Reinhold, 1990).
An elementary treatment of chaotic solutions of difference equations is:
Devaney, R. L., Chaos, Fractals, and Dynamics (Reading, MA: Addison-Wesley, 1990).
Second Order Linear Equations
Linear equations of second order are of crucial importance in the study of differential equations for two main reasons. The first is that linear equations have a rich theoretical structure that underlies a number of systematic methods of solution. Further, a substantial portion of this structure and these methods are understandable at a fairly elementary mathematical level. In order to present the key ideas in the simplest possible context, we describe them in this chapter for second order equations. Another reason to study second order linear equations is that they are vital to any serious investigation of the classical areas of mathematical physics. One cannot go very far in the development of fluid mechanics, heat conduction, wave motion, or electromagnetic phenomena without finding it necessary to solve second order linear differential equations. As an example, we discuss the oscillations of some basic mechanical and electrical systems at the end of the chapter.
3.1 Homogeneous Equations with Constant Coefficients
where f is some given function. Usually, we will denote the independent variable by t since time is often the independent variable in physical problems, but sometimes we
A second order ordinary differential equation has the form
Chapter 3. Second Order Linear Equations
will use x instead. We will use y, or occasionally some other letter, to designate the dependent variable. Equation (1) is said to be linear if the function f has the form
f(t y dd")=s(t) - p(t) d- q (t ]y (2)
that is, if f is linear in y and y'. In Eq. (2) g, p, and q are specified functions of the independent variable t but do not depend on y. In this case we usually rewrite Eq. (1) as
y"+ p(t) y' + q (t) y = g(t) (3)
where the primes denote differentiation with respect to t. Instead of Eq. (3), we often see the equation
P (t) y" + Q(t) y + R(t) y = G (t). (4)
Of course, if P (t) = 0, we can divide Eq. (4) by P (t) and thereby obtain Eq. (3) with
()- M RJH ()- Gt) (5)
p() p(t) q() p(t) g() p(t)? ()
In discussing Eq. (3) and in trying to solve it, we will restrict ourselves to intervals in which p, q, and g are continuous functions.1
If Eq. (1) is not of the form (3) or (4), then it is called nonlinear. Analytical investigations of nonlinear equations are relatively difficult, so we will have little to say about them in this book. Numerical or geometical approaches are often more appropriate, and these are discussed in Chapters 8 and 9. In addition, there are two special types of second order nonlinear equations that can be solved by a change of variables that reduces them to first order equations. This procedure is outlined in Problems 28 through 43.