Page 51 - Schaum's Outline of Differential Equations

34 EXACT FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 5

5.4. Determine whether the differential equation

is exact.
Here M(x, y) = x + sin y and N(x, y) = x cos y — 2y. Thus, dMIdy = dNIdx = cos y, and the differential equation
is exact.

5.5. Solve the differential equation given in Problem 5.4.
This equation was shown to be exact. We now seek a function g(x, y) that satisfies (5.4) and (5.5). Substituting
M(x, y) into (5.4), we obtain dgldx = x + sin y. Integrating both sides of this equation with respect to x, we find

or

To find h(y), we differentiate (1) with respect to y, yielding dgldy = x cos y + h'(y), and then substitute this
result along with N(x, y) = x cos y - 2y into (5.5). Thus we find

2
from which it follows that h(y) = —y + c^ Substituting this h(y) into (1), we obtain

The solution of the differential equation is given implicitly by (5.6) as

5.6. Solve

Rewriting this equation in differential form, we obtain

Here, and and, since the differential equation is
xy
exact. Substituting M(x, y) into (5.4), we find dgldx = 2 + ye ; then integrating with respect to x, we obtain

or
xy
To find h(y), first differentiate (1) with respect to y, obtaining dgldy = xe + h'(y); then substitute this result
along with N(x, y) into (5.5) to obtain

2
It follows that h(y) = -y + c 1. Substituting this h(y) into (1), we obtain

The solution to the differential equation is given implicitly by (5.6) as