ABSTRACT
Working with the equation in standard form, we now seek a factor m (x) with
the property that when the differential equation is multiplied by m to give
m dy
dx mp(x)ymq(x),
the left-hand side can be written as
d
dx (my),
thus reducing the equation to the simple form
d
dx (my)mq(x):
This result can be integrated to give
g ddx (my) dxgmq(x) dx, so
myAgmq(x) dx,
tion in the form
y A
m
m gmq(x) dx: It now remains for us to determine the factor m. The function m was
required to be such that
d
dx (my)m
dy
dx mp(x)y,
so performing the differentiation on the left-hand side gives
m dy
dx y
dm
dx m
dy
dx mp(x)y,
and after simplification this becomes
dm
dx mp(x):
This is a simple differential equation for m with separable variables, so
g dmm g p(x) dx and thus
ln½m½ ln Cg p(x) dx, where ln C is an arbitrary constant. It follows from this that
Cmexp g p(x) dx
:
For convenience we set C/1. This is permissible because the factor m multiplies the entire equation so the value of C is immaterial. The factor
m (x ), called the integrating factor for the first order linear differential equa-
tion in standard form, is thus
mexp g p(x) dx
:
Integration of a first order linear differential equation
To find the general solution of the first order linear differential equation
a(x) dy
dx b(x)yc(x)
dy
dx p(x)yq(x),
with p(x )/b(x )/a(x ) and q (x)/c (x )/a (x). 2 Find the integrating factor
mexp g p(x) dx
:
3 Replace the differential equation in standard form by the equivalent
differential equation
d
dx (my)mq(x):
4 Integrate the result in step 3 to obtain
myAgmq(x) dx, where A is an arbitrary constant.
