Lecture Notes in Differential Equations - Bruce E. Shapiro

Lesson 10
Integrating Factors
Definition 10.1. An integrating factor for the differential equation
is a any function such µ(t, y) such that
is exact.
M(t, y)dt + N(t, y)dy = 0 (10.1)
µ(t, y)M(t, y)dt + µ(t, y)N(t, y)dy = 0 (10.2)
If (10.2) is exact, then by theorem 9.1
∂
∂y (µ(t, y)M(t, y)) = ∂ (µ(t, y)N(t, y)) (10.3)
∂t
In this section we will discuss some special cases in which we can solve (10.3)
to find an integrating factor µ(t, y) that satisfies (10.3). First we give an
example to that demonstrates how we can use an integrating factor.
Example 10.1. Show that
( ) ( sin y
cos y + 2e
− 2e −t −t )
cos t
sin t dt +
dy = 0 (10.4)
y
y
is not exact, and then show that
µ(t, y) = ye t (10.5)
is an integrating factor for (10.4), and then use the integrating factor to
find a general solution of (10.4).
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