15.4 Linear equations and integrating factors
This class of DEs is absolutely fundamental. Such equations occur throughout science and
engineering. Alinear equation of the first orderis one that can be put in the form
dy
dx
+P(x)y=Q(x) ( 15. 3 )
whereP(x)andQ(x)are functions ofx. It is called ‘linear’ because the non-derivative
part is linear in the dependent variabley:
dy
dx
=−P(x)y+Q(x) (cf:ay+b)
This is key to the method of solution – if any other power ofybut 0 or 1 occurred
then what we are about to do would not be possible. We notice that the left-hand side of
equation (15.3) looks very much like the derivative of a product:
dy
dx
+Py
compared to say
d(uy)
dx
=u
dy
dx
+
du
dx
y
The resemblance can be improved if we multiply through by a function ofx,I=I(x),
yet to be determined (called anintegrating factorbecause it enables us to integrate the
equation). So, compare
I
dy
dx
+IPy=IQ
with
d
dx
(Iy)=I
dy
dx
+
dI
dx
y
NowIQis a function ofxalone, with noy. It can therefore be integrated with respect
tox. On the other hand, if we now takeIto be such that
dI
dx
=IP
then
I
dy
dx
+IPy=I
dy
dx
+
dI
dx
y
=
d
dx
(Iy)
and we have
d(Iy)
dx
=IQ