7.2 Some special types of DE
7.2.1 Separable type
Equations of the form
are called separable. They are dealt with in the
following way: Divide both sides by
, and integrate both sides with respect to
,
Now do both integrals.
Example 7.3:
-
Solve the DE
, given that
when
.
Solution:
-
Divide by
, and obtain
Now integrate both sides with respect to
This is the general solution, but we know
that at
,
. Substituting this we find that
, therefore
and
Example 7.4:
-
Find the general solution of
Solution:
-
Rearrange as
So here
,
. Divide by
,
Integrate both sides with respect to
We write
, with
also arbitrary, but positive. We find
Thus
, or isolating
,
Example 7.5:
-
satisfies the DE
Given that
at
find
at
.
Solution:
-
Here
, i.e., a constant, and
, so
Since at
,
, we have
, and
At
,
.
7.2.2 linear type
These have form,
| (7.1) |
Method as follows
N.B. Remember the method
not the final formula!
Example 7.6:
-
Find the general solution of
| (7.4) |
Solution:
-
Here
so
(no constant of integration needed here),
. Multiply both sides of (
7.4
) by
:
The l.h.s. is the differential of
so we find
Integrate this and find
Thus, finally,
Example 7.7:
-
Solve the DE
| (7.5) |
given that
when
.
Solution:
-
Rearrange (
7.5
) ,
which is of linear form with
. We find
and
. Multiply (
7.6
) by
, and find
the l.h.s. is differential of
. Integrate this and find
This is the general solution. We know that
when
then
so
. Therefore
and
7.2.3 Homogeneous Type
We first need to define a
function of two variables:
If
is a function of 2 variables, it delivers a number on
specification of
and
.
Examples:
,
,
.
If
and
in the above we get
,
,
.
Now we can define a homogeneous
function:
A homogeneous function of 2 variables is
one where we have a sum of terms all of which have the same total
power (called degree).
Examples
function | degree |
|
|
| 2 |
| 1 |
| 1 |
| 0 |
| 2 |
| not homogeneous |
| not homogeneous |
|
|
|
There is a simple test to see if
is homogeneous. Replace
by
and
by
to get
. If
then
is homogeneous with degree
.
Example 7.8:
-
-
:
and
is homogeneous with degree 1.
-
:
which is therefore homogeneous of degree
0.
-
.
which is therefore homogeneous of degree
0.
A homogeneous DE is one of type
, with
and
both homogeneous and of the same degree.
Homogeneous DEs can be made separable by the substitution
. We shall demonstrate this by means of examples:
Example 7.9:
-
Find general solution of
Solution:
-
Put
, then
, so
therefore
which is separable. This can be solved in
the standard way,
And we conclude that
(We can also replace
with
.)
Often we need to rearrange the equation
first to get a homogeneous form, as in the following
example.
Example 7.10:
-
Solve
given
when
.
Solution:
-
Rearrange as
This is therefore a homogeneous DE. We
substitute
,
We can now turn the crank,
which is the general solution. Imposing the
condition that for
,
, we obtain
, and therefore
. The solution is thus