Chapter 47
Homogeneous first order
differential equations
47.1 Introduction
Certain first order differential equations are not of the
‘variable-separable’ type, but can be made separable by
changing the variable.
An equation of the formP
dy
dx
=Q,wherePandQare
functions of bothxandyof the same degree throughout,
is said to behomogeneousinyand x.Forexam-
ple,f(x,y)=x^2 + 3 xy+y^2 is a homogeneous function
since each of the three terms are of degree 2. However,
f(x,y)=
x^2 −y
2 x^2 +y^2
is not homogeneous since the term
inyin the numerator is of degree 1 and the other three
terms are of degree 2.
47.2 Procedure to solve differential
equations of the formP
dy
dx
=Q
(i) RearrangeP
dy
dx
=Qinto the form
dy
dx
=
Q
P
.
(ii) Make the substitutiony=vx(wherevis a func-
tion ofx), from which,
dy
dx
=v( 1 )+x
dv
dx
,bythe
product rule.
(iii) Substitute for bothy and
dy
dx
in the equation
dy
dx
=
Q
P
. Simplify, by cancelling, and an equation
results in which the variables are separable.
(iv) Separate the variables and solve using the method
shown in Chapter 46.
(v) Substitutev=
y
x
to solve in terms of the original
variables.
47.3 Worked problemson
homogeneous first order
differential equations
Problem 1. Solve the differential equation:
y−x=x
dy
dx
,givenx=1wheny=2.
Using the above procedure:
(i) Rearrangingy−x=x
dy
dx
gives:
dy
dx
=
y−x
x
,
which is homogeneous inxandy.
(ii) Lety=vx,then
dy
dx
=v+x
dv
dx
(iii) Substituting foryand
dy
dx
gives:
v+x
dv
dx
=
vx−x
x
=
x(v− 1 )
x
=v− 1