General Contents
Detailed Contents
Index
Reduction of Order in a Differential Equation
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Given that
is a solution of
,
find the second independent solution,
y
2
(
x
).
General Contents
Detailed Contents
Index
What technique can be used here?
We can use Reduction of Order.
Express
y
2
in terms of
y
1
for this method.
We use another function,
v
:
y
2
=
v
*
y
1
Will we need derivatives of
v
?
Yes.
Substitute the expression for
y
1
into
y
2
.
We get
What do we do next?
We need to find
and substitute into the original differential equation,
.
What rules of differentiation do we need for finding
y
2
'?
To differentiate
, we need the
Product Rule
and
Power Rule
.
Apply the Product Rule to set up finding
y
2
'.
Complete the differentiation.
We get
or in the usual order we have
Use the Product and Power Rules again to set up finding
y
''.
Complete the differentiation.
Collect like terms and write them in the usual order for derivatives.
Substitute these results for
into our original differential equation,
We get
Distribute the multiplication.
Collect like terms.
We get
What is the order of this differential equation?
Second order.
What order do we need in order to be able to solve a differential equation?
So far, we know how to solve only first order differential equations.
How can we convert to a first order equation?
We can define a new function,
w
=
v
'.
We will also need the first derivative of
w
.
Find
w
'.
Since
Substitute these results in our differential equation for
v
:
We get
Is this a first order differential equation?
Yes.
How can we solve it?
We can separate variables.
Set this up.
Separate the variables.
What next?
We can integrate both sides.
Set that up.
What is the result on the left-hand side?
What is the result on the right-hand side?
Why do we use ln
C
?
We need a constant, and we can write it in this form as well.
It will make simplification easier later.
How do we solve for
w
?
We use each side as an exponent with e as a base.
Do it.
We get
Simplify the left-hand side.
Simplify the right-hand side using
Properties of Logarithms
.
So we have
Use the definition of
w
=
v
' to convert to a differential equation involving
v
.
We get
.
How do we solve this?
Again, we can separate variables.
Set that up.
Do the integration.
We get
Now use the definition of
y
2
=
v * x
-1
to finally get
y
2
.
Check this by determining
y
2
' and
y
2
'' from this result for
y
2
and substitute in the original differential equation.
First, what rule of differentiation should we use to find
y
2
'?
The Power Rule.
Use it to find
y
2
'.
Use the Power Rule again to find
y
2
''.
Substitute these results into the original differential equation,
We get
Distribute the multiplication and collect the like terms.
We get
It checks!
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