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 form P
dy
dx
= Q, where P and Q are
functions of both x and y of the same degree throughout,
is said to be homogeneous in y and x. For example, f (x, y) = x 2 + 3x y + y 2 is a homogeneous function
since each of the three terms are of degree 2. However,
f (x, y) =
x 2 − y
2x 2 + y 2 is not homogeneous since the term
in y in the numerator is of degree 1 and the other three
terms are of degree 2.
47.2 Procedure to solve differential
equations of the form P
dy
dx
= Q
(i) Rearrange P
d y
dx
= Q into the form
d y
dx
=
Q
P
.
(ii) Make the substitution y = vx (where v is a function of x), from which,
d y
dx
= v(1) + x
dv
dx
, by the
product rule.
(iii) Substitute for both y and
d y
dx
in the equation
d y
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) Substitute v =
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
d y
dx
, given x = 1 when y = 2.
Using the above procedure:
(i) Rearranging y − x = x
d y
dx
gives:
d y
dx
=
y − x
x
,
which is homogeneous in x and y.
(ii) Let y = vx, then
d y
dx
= v + x
dv
dx
(iii) Substituting for y and
d y
dx
gives:
v + x
dv
dx
=
vx − x
x
=
x(v − 1)
x
= v − 1
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 form P
dy
dx
= Q, where P and Q are
functions of both x and y of the same degree throughout,
is said to be homogeneous in y and x. For example, f (x, y) = x 2 + 3x y + y 2 is a homogeneous function
since each of the three terms are of degree 2. However,
f (x, y) =
x 2 − y
2x 2 + y 2 is not homogeneous since the term
in y in the numerator is of degree 1 and the other three
terms are of degree 2.
47.2 Procedure to solve differential
equations of the form P
dy
dx
= Q
(i) Rearrange P
d y
dx
= Q into the form
d y
dx
=
Q
P
.
(ii) Make the substitution y = vx (where v is a function of x), from which,
d y
dx
= v(1) + x
dv
dx
, by the
product rule.
(iii) Substitute for both y and
d y
dx
in the equation
d y
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) Substitute v =
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
d y
dx
, given x = 1 when y = 2.
Using the above procedure:
(i) Rearranging y − x = x
d y
dx
gives:
d y
dx
=
y − x
x
,
which is homogeneous in x and y.
(ii) Let y = vx, then
d y
dx
= v + x
dv
dx
(iii) Substituting for y and
d y
dx
gives:
v + x
dv
dx
=
vx − x
x
=
x(v − 1)
x
= v − 1