Chapter 48
Linear first order differential
equations
48.1 Introduction
An equation of the form
d y
dx
+ P y = Q, where P and
Q are functions of x only is called a linear differential equation since y and its derivatives are of the first
degree.
(i) The solution of
d y
dx
+ P y = Q is obtained by
multiplying throughout by what is termed an
integrating factor.
(ii) Multiplying
d y
dx
+ P y = Q by say R, a function
of x only, gives:
R
d y
dx
+ RPy = RQ
(1)
(iii) The differential coefficient of a product Ry is
obtained using the product rule,
i.e.
d
dx
(Ry) = R
d y
dx
+ y
d R
dx
,
which is the same as the left hand side of
equation (1), when R is chosen such that
RP =
dR
dx
(iv) If
dR
dx
= RP, then separating the variables
gives
dR
R
= P dx.
Integrating both sides gives:
dR
R
=
P dx i.e. ln R =
P dx + c
from which,
R = e
P dx+c
= e
P dx e
c
i.e. R = Ae
P dx , where A = e c = a constant.
(v) Substituting R = Ae
P dx in equation (1) gives:
Ae
P dx
d y
dx
+ Ae
P dx P y = Ae
P dx Q
i.e. e
P dx
dy
dx
+ e
P dx P y = e
P dx Q (2)
(vi) The left hand side of equation (2) is
d
dx
ye
P dx
which may be checked by differentiating
ye
P dx with respect to x, using the product rule.
(vii) From equation (2),
d
dx
ye
P dx
= e
P dx Q
Integrating both sides gives:
ye
P dx
=
e
P dx Q dx
(3)
(viii) e
P dx is the integrating factor.