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