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Linear Nonhomogeneous Differential Equations and Variation of Parameters: Example 4
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Solve
.
What type of differential equation is this?
It is a linear nonhomogeneous differential equation with constant coefficients.
How can we solve it?
We can use the
Method of Variation of Parameters
.
How do we start?
We find
, the homogeneous solution, which takes care of the case of 0 on the right-hand side, and
, a particular solution, which handles the actual right-hand side.
How do we find the homogeneous solution?
Since
must satisfy
, we can see that any function for which the first and third derivatives contain the same function with opposite signs can be a term in
.
Can a constant
be a solution?
Yes, because its first and third derivatives are zero.
Can
be a solution?
Yes, because its first and third derivatives both involve the sine function with opposite signs.
Can
be a solution?
Yes, because its first and third derivatives both involve the cosine function with opposite sign.
Combine these results to get the homogeneous solution .
We get
. This is the
Homogeneous Solution
.
What is the form of the particular solution?
We use the same functions as are in the homogeneous solution. However, each is multiplied by a parameter instead of a constant.
Let’s set that up in terms of
from the
Method of Variation of Parameters
.
How many terms do we need?
Three, because of the three terms in
Write the general form of the particular solution.
What are
here?
We use
from the homogeneous solution.
Substitute these into the particular solution.
What do
represent?
They are functions of
x
.
How do we determine them?
We use the equations from
Method of Variation of Parameters
How many equations are needed?
Three, in order to determine our three functions,
Which equations do we use from step 3 of the Method of Variation of Parameters?
We use the first two and the last equations.
What is the value of
n
in the last equation?
It is the same as the number of terms in our particular solution.
n
= 3 in this case.
What is
here?
Write the first equation using these results.
Eq. 1.
Set up the second equation.
Eq. 2.
Set up the third equation.
Using
n
= 3, we get for the last equation in
Method of Variation of Parameters
.
Set up the second derivatives of the
’s and substitute for
.
Using the first derivative from Equation 2, we get
Eq. 3.
So we have 3 equations in the 3 unknowns
. How do we solve for them?
We can combine Eq. 2 and Eq. 3. Do that.
We multiply Equation 2 by cos(
x
) and Equation 3 by sin(
x
) and subtract. First, do the multiplications.
Now subtract.
We get
Use trig identities to simplify.
Now use the above result for
in Equation 2 to find
.
We get
Solve for
.
Use these two results in Equation 1 to find
We get
We now have the derivatives of
. Using a table of integrals, solve for
.
Determine
Solve for
Combine the results for
to get
.
This is the
particular solution.
Combine this with the homogeneous solution to get the full solution.
This is the
full solution.
The end.
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General Contents
Detailed Contents
Index
.