11.3 Separable equations 321
The differential equation
(11.16)
is made separable by means of the substitutionu 1 = 1 y 2 x. Then
(11.17)
so that, separating the variables xand u,
(11.18)
and the solution of the equation is
(11.19)
An equation of this type is often called a homogeneous equation because a function
of type (11.14) is a homogeneous function. In general, a function of one or more
variables is a homogeneous function of degree nif
f(λx, λy, =) 1 = 1 λ
nf(x,y, =) (11.20)
The function (11.14) is therefore homogeneous of zero degree.
EXAMPLE 11.5Solve.
The substitutiony 1 = 1 xugives
so that
.
Then, separating the variables and integrating,
ZZudu
dx
x
u
=, =+xc
22
ln
x
du
dx u
=
1
dy
dx
ux
du
dx
u
u
=+ =+
1
dy
dx
xy
xy
=
22Z
du
fu u
xc
()
ln
−
=+
du
fu u
dx
()− x
=
yxu
dy
dx
ux
du
dx
==+=,(fu)
dy
dx
Fxy f
y
x
=,=
()