Chapter 48
Linear first order differential
equations
48.1 Introduction
An equation of the form
dy
dx
+Py=Q,wherePand
Qare functions ofxonly is called alinear differen-
tial equationsinceyand its derivatives are of the first
degree.
(i) The solution of
dy
dx
+Py=Q is obtained by
multiplying throughout by what is termed an
integrating factor.
(ii) Multiplying
dy
dx
+Py=Qby sayR, a function
ofxonly, gives:
R
dy
dx
+RPy=RQ (1)
(iii) The differential coefficient of a productRyis
obtained using the product rule,
i.e.
d
dx
(Ry)=R
dy
dx
+y
dR
dx
,
which is the same as the left hand side of
equation (1), whenRis chosen such that
RP=
dR
dx
(iv) If
dR
dx
=RP, then separating the variables
gives
dR
R
=Pdx.
Integrating both sides gives:
∫
dR
R
=
∫
Pdxi.e. lnR=
∫
Pdx+c
from which,
R=e
∫
Pdx+c=e
∫
Pdxec
i.e.R=Ae
∫
Pdx,whereA=ec=a constant.
(v) SubstitutingR=Ae
∫
Pdxin equation (1) gives:
Ae
∫
Pdx
(
dy
dx
)
+Ae
∫
PdxPy=Ae
∫
PdxQ
i.e. e
∫
Pdx
(
dy
dx
)
+e
∫
PdxPy=e
∫
PdxQ (2)
(vi) The left hand side of equation (2) is
d
dx
(
ye
∫
Pdx
)
which may be checked by differentiating
ye
∫
Pdxwithrespect tox, usingthe product rule.
(vii) From equation (2),
d
dx
(
ye
∫
Pdx
)
=e
∫
PdxQ
Integrating both sides gives:
ye
∫
Pdx=
∫
e
∫
PdxQdx (3)
(viii) e
∫
Pdxis theintegrating factor.