Separable Equations
We have found a strategy for solving firstorder linear equations, and would now like to consider a
particular class of firstorder nonlinear equations. We will again make use of exact derivatives.
Definition
A differential equation that can be put in the form
dy
dx
=
g
(
x
)
f
(
y
)
is called
separable
.
To solve separable equations, we can multiply by
f
(
y
)
to get
(1)
f
(
y
)
dy
dx
=
g
(
x
)
and observe that the LHS looks like the result of applying the chain rule to some function
F
(
y
)
where
F
is an antiderivative for
f
. In particular, we have from the chain rule
d
dx
F
(
y
)
=
F
0
(
y
)
dy
dx
=
f
(
y
)
dy
dx
.
So we can rewrite the LHS of (1) as an exact derivative to get
d
dx
F
(
y
)
=
g
(
x
)
and integrating both sides gives
F
(
y
) =
Z
g
(
x
)
dx.
If
G
is an antiderivative for
g
then we have
(2)
F
(
y
) =
G
(
x
) +
C.
Now from equation (2), we may or may not be able to solve for
y
as an explicit function of
x
. But
even if we can’t, equation (2) still defines
y
as an implicit function of
x
, and is considered to be the
general solution of the differential equation. In this case (2) is called the set of
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 '08
 staff
 Linear Equations, Equations, Derivative, Quadratic equation, dx, dy

Click to edit the document details