Page 64 - Schaum's Outline of Differential Equations

CHAP. 6] LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 47

Substituting these equations into the differential equation, we obtain

x /2
This last equation is linear. Its solution is found in Problem 6.10 to be z = ce +1. The solution of the original
differential equation is then

6.17. Solve

4
This is a Bernoulli differential equation wi\hp(x) = -3/x, q(x) = x , and n = ^. Using Eq. (6.5), we make the
1
1/2
3 2
213
substitution z = y ~ (1/3) = y . Thus, y = z ' and y = |z z'. Substituting these values into the differential equation,
we obtain

5
213
This last equation is linear. Its solution is found in Problem 6.12 to be z = ex 2 + ^x . Since z = y , the solution of
5
2 3
2
5 312
the original problem is given implicitly by y ' = ex + ^x , or explicitly by y = ± (ex 2 + ^x ) .
6.18. Show that the integrating factor found in Problem 6.1 is also an integrating factor as defined in Chapter 5
Eq. (5.7).
The differential equation of Problem 6.1 can be rewritten as

which has the differential form

or

3x
Multiplying (1) by the integrating factor I(x) = e , we obtain

Setting

we have

from which we conclude that (2) is an exact differential equation.

6.19. Find the general form of the solution of Eq. (6.1).
Multiplying (6.1) by (6.2), we have

Since