Higher Engineering Mathematics

I

Differential equations

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, wherePand

Qare functions of bothxandyof the same degree
throughout, is said to behomogeneousinyandx.
For example,f(x,y)=x^2 + 3 xy+y^2 is a homoge-
neous 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 homo-

geneous 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

, by the

product rule.

(iii) Substitute for bothy and

dy

dx

in the equa-

tion

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 problems on

homogeneous first order

differential equations

Problem 1. Solve the differential equation:

y−x=x

dy

dx

,givenx=1 wheny=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

(iv) Separating the variables gives:

x

dv

dx

=v− 1 −v=−1, i.e.dv=−

1

x

dx

Integrating both sides gives:

∫

dv=

∫

−

1

x

dx

Hence,v=−lnx+c

(v) Replacingvby

y

x

gives:

y

x

=−lnx+c, which

is the general solution.

Whenx=1,y=2, thus:

2

1

=−ln 1+cfrom

which,c= 2