General Contents
Detailed Contents
Index
Reduction of Order in a Differential Equation, y not Present: Example 1
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Solve
Equation 1
.
What is different about this differential equation?
There is a second derivative present, and
y
itself 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
What type of differential equation is this?
It is a first order linear differential equation.
What is the general form of this type of equation?
What is the general solution?
General Solution:
Equation 3
What is P(x) in this problem?
Express
for this problem.
How do we proceed?
We need to determine
, substitute it into Equation 3, and then do the integral on the right-hand side.
Determine
Determine
We get
Substitute this result and
into Equation 3.
Equation 3 becomes
Integrate by parts, or use #73 from the Table of Integrals.
We get
Equation 4
:
How can we solve for
w
?
We need to multiply by
on both sides of the equation.
Do that.
We want a solution involving
y
.
How can we get it from this result?
We can substitute the definition of
w
into equation 4..
Do that.
We get
How can we solve this differential equation?
We can use separation of variables.
Rewrite the equation with the variables separated.
Do the integration.
We get
, or
, or
Equation 5
How can we check this result?
We can determine
from equation 5 and substitute into equation 1.
Determine
.
Determine
.
Substitute into equation 1.
We get
.
Distribute the multiplication and collect like terms.
We get
It checks!
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General Contents
Detailed Contents
Index
Equation 1
.