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.