 
Summary: 18.03ESG Notes 1
Pramod N. Achar
Fall 1999
Yesterday in class, the question arose of how one might derive the integrating
factor to use in solving a firstorder linear equation:
dy
dx
+ P(x)y = Q(x). (1)
We want to multiply through by some (x) such that the lefthand side becomes
recognizable as the derivative of something. (The righthand side stays just a
function of x, so we can just integrate it.)
(x)
dy
dx
+ P(x)(x)y = Q(x)(x) (2)
More specifically, we are going to try to obtain the lefthand side as the result
of using the product rule for differentiation. Looking at the two terms on the
lefthand side, we see that one contains dy/dx and the other contains y, so it is
reasonable to suppose that y ought to be one of the factors in the product we're
looking for. Moreover, everything else on the lefthand side is just a function of
