312 CHAPTER 5 Series and Differential Equations
5.5.3 Linear first-order ODE; integrating factors
In this subsection we shall consider the general first-order linear ODE:
y′+p(x)y = q(x). (5.3)As we’ll see momentarily, these are, in principle, very easy to solve.
The trick is to multiply both sides of (5.3) by theintegrating factor
μ(x) = e∫
p(x)dx.Notice first thatμ(x) satisfiesμ′(x) =p(x)μ(x). Therefore if we multi-
ply (5.3) through byμ(x) we infer that
d
dx(μ(x)y) =μ(x)y′+p(x)μ(x)y=μ(x)q(x),from which we may conclude that
μ(x)y =∫
μ(x)q(x)dx.Example 1. Find the general solution of the first-order ODE
(x+ 1)y′−y=x, x >− 1.First of all, in order to put this into the general form (5.3) we must
divide everything byx+ 1:
y′−1
x+ 1y=x
x+ 1.
This implies that an integrating factor is
μ(x) = e−∫dx
x+1 =^1
x+ 1.
Multiply through byμ(x) and get