Equations of the form
|
are called separable. They are dealt with in the following way: Divide both sides by
Now do both integrals.
Example 7.3:
Solve the DE
, given that
Solution:
Divide by
Now integrate both sides with respect to
This is the general solution, but we know that at
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Example 7.4:
Find the general solution of
Solution:
Rearrange as
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So here
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Integrate both sides with respect to
We write
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Thus
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Example 7.5:
Given that
Solution:
Here
Since at
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At
These have form,
| (7.1) |
Method as follows
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and thus
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Since
![]() | (7.2) |
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Note also that
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This is exactly the l.h.s. of (7.2). Rewrite eq. (7.2) as
![]() | (7.3) |
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Hence
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N.B. Remember the method not the final formula!
Example 7.6:
Solution:
Here
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The l.h.s. is the differential of
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Integrate this and find
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Thus, finally,
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Example 7.7:
Solution:
Rearrange (7.5) ,
![]() | (7.6) |
which is of linear form with
and
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the l.h.s. is differential of
This is the general solution. We know that when
. Therefore
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We first need to define a function of two variables:
function | degree |
2 | |
1 | |
1 | |
0 | |
2 | |
not homogeneous | |
not homogeneous | |
There is a simple test to see if y)
λy)
λy)=λnf(x
y)
Example 7.8:
![]() ![]() |
and
which is therefore homogeneous of degree 0.
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which is therefore homogeneous of degree 0.
A homogeneous DE is one of type y)g(x
y)
Example 7.9:
Find general solution of
Solution:
Put
therefore
which is separable. This can be solved in the standard way,
And we conclude that
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(We can also replace
Often we need to rearrange the equation first to get a homogeneous form, as in the following example.
Example 7.10:
Solve
given
Solution:
Rearrange as
This is therefore a homogeneous DE. We substitute
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We can now turn the crank,
which is the general solution. Imposing the condition that for
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