Page 37 - Advanced engineering mathematics

1.2 Linear Equations 17

This is the general solution with the arbitrary constant c.
We do not recommend memorizing this formula for y(x). Instead, carry out the following
procedure.

Step 1. If the differential equation is linear, y + p(x)y = q(x). First compute

e p(x)dx .
This is called an integrating factor for the linear equation.
Step 2. Multiply the differential equation by the integrating factor.
Step 3. Write the left side of the resulting equation as the derivative of the product of y and the
integrating factor. The integrating factor is designed to make this possible. The right side
is a function of just x.
Step 4. Integrate both sides of this equation and solve the resulting equation for y, obtain-
ing the general solution. The resulting general solution may involve integrals (such as
2
cos(x )dx) which cannot be evaluated in elementary form.

EXAMPLE 1.8
The equation y + y = x is linear with p(x) = 1 and q(x) = x. An integrating factor is

p(x)dx dx x
e = e = e .
Multiply the differential equation by e to get
x
x
x
x

e y + e y = xe .
This is
x
(ye ) = xe x
with the left side as a derivative. Integrate this equation to obtain

x x x x
ye = xe dx = xe − e + c.
−x
Finally, solve for y by multiplying this equation by e :
−x
y = x − 1 + ce .
This is the general solution, containing one arbitrary constant.

EXAMPLE 1.9
Solve the initial value problem
y
2

y = 3x − ; y(1) = 5.
x
This differential equation is not linear. Write it as
1
2
y + y = 3x ,
x
which is linear. An integrating factor is
(1/x)dx ln(x)
e = e = x
for x > 0. Multiply the differential equation by x to obtain
xy + y = 3x 3

or
3

(xy) = 3x .

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-17 27410_01_ch01_p01-42