Page 49 - Schaum's Outline of Differential Equations

32 EXACT FIRST-ORDER DIFFERENTIAL EQUATIONS [CHAR 5

INTEGRATING FACTORS
In general, Eq. (5.7) is not exact. Occasionally, it is possible to transform (5.7) into an exact differential
equation by a judicious multiplication. A function I(x, y) is an integrating factor for (5.7) if the equation

is exact. A solution to (5.7) is obtained by solving the exact differential equation defined by (5.7). Some of the
more common integrating factors are displayed in Table 5-1 and the conditions that follow:

a function of x alone, then

a function of y alone, then

Table 5-1

Group of terms Integrating factor I(x, y) Exact differential dg (x, y)

y dx-x dy

y dx-x dy

y dx-x dy

y dx-x dy

y dx + x dy

y dx + x dy

y dy + x dx

y dy + x dx

ay dx + bx dy
(a, b constants)