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Reduction of Order in a Differential Equation, x not Present: Example 2
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Solve
Equation 1
.
What is different about this differential equation?
There is a second derivative present, and
x
is absent.
What is the order of this differential equation?
Because of the second derivative, it is called a Second Order differential equation.
Can we use the previous methods?
No, they are limited to first order differential equations.
What can we do?
Since
is the derivative of
, we can reduce the order via a substitution.
Using
w
as a new variable, what is that substitution?
We let
Determine
in terms of
w
We get
Rewrite Equation 1 using
.
The original differential equation,
becomes
.
Equation 2
Can we separate variables here?
No, we need to reduce it to 2 variables.
How could we change to a dependence on
?
We could use the chain rule on the derivative.
Do that.
We get
.
How does this help?
We can use our definition
to get
Use these results to rewrite equation 2.
Equation 3
How is this equation better than equation 2?
It is expressed in terms of
.
How do we proceed?
Let’s solve for
so that we can check whether this is a separable differential equation.
Isolate
.
Multiply both sides by
We get
Is it separable?
Yes.
Separate the variables.
Do the integration.
, or
Solve for
w
.
We get
Equation 4
How can we go further?
We can replace
w
in equation 4 by its definition.
Do that.
We get
Is this equation separable?
Yes.
Separate the variables
Do the integration.
, or
Equation 5
Can we go any further?
No.
How can we check this result?
We can determine
from equation 5 and substitute into equation 1.
Determine
.
Differentiating implicitly, we get
Solve for
Determine
.
Replace
in this expression.
Substitute into equation 1.
We get
. It checks.!
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General Contents
Detailed Contents
Index
Equation 1
.