Homogeneous Differential Equations: Example 3
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Show that the differential equation,
,
is homogeneous, and obtain its general solution.
How do we show that it is homogeneous?
One way is to substitute
tx
for
x
and
ty
for
y
, and then compare the resulting equation with the original equation.
If the equation is homogeneous, what do we get for the comparison?
The new equation will be
t
n
times the original.
We use the phrase “homogeneous of degree ___.”
What is the degree here?
The exponent of t, or n.
Substitute
tx
for
x
and
ty
for
y
in the original equation.
We get
Can we factor
t
outside the radical?
Yes.
Do it.
, or
Can we factor out a power of t?
Yes.
Do it.
What is our conclusion about homogeneity?
The equation is homogeneous.
What is the degree?
One.
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
Substitute our result for
y
and
dy
in the original equation.
Distribute the multiplication.
Simplify by collecting the
vxdx
terms.
,
or factoring the radical and moving
dv
to the other side, we get
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,
using
.
Putting the constant on the left side, we get
Are we done?
No.
What do we do next?
Substitute
to get a result in terms of
y
and
x
.
Do it.
Equation 1
How can we check this result?
We could determine
from Equation 1 and compare it to the original problem,
Set that up.
From Equation 1, we get
What do we do next?
Use the Chain Rule.
Do it.
We get
Multiply each side by
to clear the denominators.
Multiply x into the radical.
Multiply by (-1) on both sides and reorder.
We get
, which checks.
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General Contents
Detailed Contents
Index