Page 46 - Advanced engineering mathematics

26 CHAPTER 1 First-Order Differential Equations

(d) Show that δ(x, y) = xy −3 is also an integrating is not exact. Solve this equation by finding an integrat-
b
a
factor on any rectangle where x = 0and y = 0. ing factor of the form μ(x, y) = x y . Hint: Consider
Use this integrating factor to find the general the differential equation
solution.
2
(e) Write the differential equation as μx y + μxy =−μy −3/2

1

y − y = 0
x and solve form a and b so that equations (1.6) are
and solve it as a linear differential equation. satisfied.
(f) How do the solutions found in parts (b) through 16. Try the strategy of Problem 15 on the differential
(e) differ from each other? equation
15. Show that
2
2
2

x y + xy =−y −3/2 2y − 9xy + (3xy − 6x )y = 0.
1.4 Homogeneous, Bernoulli, and Riccati Equations
We will discuss three other types of first-order differential equations for which techniques of
solution are available.

1.4.1 The Homogeneous Differential Equation

A homogeneous differential equation is one of the form

y = f (y/x)
with y isolated on one side and on the other an expression in which x and y always occur

2
2
in the combination y/x. Examples are y = sin(y/x) − x/y and y = x /y .

In some instances, a differential equation can be manipulated into homogeneous form. For
example, with
y

y =
x + y
we can divide numerator and denominator on the right by x to obtain the homogeneous equation
y/x

y = .
1 + y/x
This manipulation requires the assumption that x = 0.
A homogeneous differential equation can always be transformed to a separable equation by
letting

y = ux.

To see this, compute y = u x + u and write u = y/x to transform

y = u x + u = f (y/x) = f (u).
In terms of u and x,thisis

xu + u = f (u)

Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.

October 14, 2010 14:9 THM/NEIL Page-26 27410_01_ch01_p01-42