Page 50 - Schaum's Outline of Differential Equations

CHAP. 5] EXACT FIRST-ORDER DIFFERENTIAL EQUATIONS 33

If M= yf(xy) and N = xg(xy), then

In general, integrating factors are difficult to uncover. If a differential equation does not have one of the
forms given above, then a search for an integrating factor likely will not be successful, and other methods of
solution are recommended

Solved Problems

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5.1. Determine whether the differential equation 2xy dx+ (1 + x )dy = 0 is exact.
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This equation has the form of Eq. (5.1) with M(x, y) = 2xy and N(x, y) = 1 + x . Since dMIdy = dNIdx = 2x, the
differential equation is exact.

5.2. Solve the differential equation given in Problem 5.1.
This equation was shown to be exact. We now determine a function g(x, y) that satisfies Eqs. (5.4) and (5.5).
Substituting M(x, y) = 2xy into (5.4), we obtain dgldx = 2xy. Integrating both sides of this equation with respect to
x, we find

or

Note that when integrating with respect to x, the constant (with respect to x) of integration can depend on y.
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We now determine h(y). Differentiating (1) with respect to y, we obtain dgldy = x + h'(y). Substituting this
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equation along with N(x, y) = 1 + x into (5.5), we have

Integrating this last equation with respect to y, we obtain h(y) = y + Cj (cj = constant). Substituting this expression
into (1) yields

The solution to the differential equation, which is given implicitly by (5.6) as g(x, y) = c, is

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Solving for y explicitly, we obtain the solution as y = C 2l(x + 1).

5.3. Determine whether the differential equation y dx - x dy = 0 is exact.
This equation has the form of Eq. (5.1) with M(x, y) = y and N(x, y) = -x. Here

which are not equal, so the differential equation as given is not exact.