General Contents
Detailed Contents
Index
Homogeneous Differential Equations: Example 1
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Show that the differential equation,
, is homogeneous, and obtain its general solution.
General Contents
Detailed Contents
Index
How do we show that it is homogeneous?
One way is to let
, and then compare
f
(
tx,ty
) with
f
(
x,y
).
If
f
(
x
,y
) is homogeneous, what do we get for the comparison?
We use the phrase “homogeneous of degree ___.”
What is the degree here?
The exponent of t, or n.
Find
for
Can we factor out a power of t?
Yes.
Do it.
What is our conclusion about homogeneity?
f
(
x
,y
) is homogeneous.
What is the degree?
Zero.
How can we use the homogeneity to get the general solution?
We can introduce a new variable,
v
.
How is
v
defined?
We’ll need to substitute for
y
.
Express
y
in terms of
v
and
x
.
Would we use the same substitution for other degrees of homogeneity?
Yes.
We’ll also need to determine
dy
from
y
=
vx
. What rule of differentiation do we need to use?
The Product Rule.
Find
dy
.
dy
=
vdx
+
xdv
To apply this, we need to rewrite
in differential form. Do this.
, or
Substitute our result for
y
and
dy
.
Simplify by collecting the
vdx
terms.
We get
, or
moving
dv
to the other side, we get
Factor the x on the left.
Can we finally separate variables?
Yes.
How?
By dividing by
on both sides.
Do it.
, or
Are the variables separated now?
Yes.
Set up the integration.
Do the integration.
Putting the constant on the left side, we get
Combine the logs on the left.
We get
Take the inverse on both sides of the equation.
Simplify.
We get
Are we done?
No.
What do we do next?
Substitute
to get a result in terms of
y
and
x
.
Do it.
Solve for y by multiplying both sides by x.
, or
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, is homogeneous, and obtain its general solution.