General Contents
Detailed Contents
Index
Linear Nonhomogeneous Differential Equations and Variation of Parameters: Example 2
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Solve the differential equation
.
General Contents
Detailed Contents
Index
What type of differential equation is this?
It is a linear nonhomogeneous differential equation with constant coefficients.
Will the method of undetermined coefficients work here?
No.
Why?
The derivative of the right-hand side does not become zero or repetitive.
How can we solve it?
We can use the
Method of Variation of Parameters
.
How do we start?
We find
y
h
, the homogeneous solution, which takes care of the case of 0 on the right-hand side, and
y
p
, a particular solution, which handles the actual right-hand side.
How do we find the homogeneous solution?
We find the
characteristic equation
.
Set up that equation.
Solve for
m
.
Factor to get
So
m
= ±
i
.
How do we use this result?
We get
This is the
Homogeneous Solution.
What method can be used here to get the particular solution?
The
Method of Variation of Parameters
.
What is the form of the particular solution here?
We use the same functions as are in the homogeneous solution.
Each is multiplied by a parameter.
Set that up in terms of
y
i
How many of these do we need?
Two, because of the two terms in
y
h
.
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 the
Method of Variation of Parameters
.
How many equations are needed?
Two, for our two functions.
Which equations do we use?
We use the first and the last equations.
What is the value of
n
?
It is the same as the number of terms in our particular solution.
n
= 2 in this case.
What is
F(x)
here?
Write the first equation using these results.
Eq. 1.
Set up the second equation.
Using n = 2, we get for the last equation in the
Method of Variation of Parameters
Set up the derivatives of the
y
i
’s and substitute for
F(x)
.
We get
Do the derivatives.
Using the product rule in the second term, we get
Eq. 2.
So we have 2 equations in the 2 unknowns
.
How do we solve for them?
Either
the Elimination Method or the Substitution Method.
Let’s try substitution.
Solve
Eq. 1.
for
Substitute this in
Eq. 2.
Do the multiplication.
Simplify by multiplying both sides by sin
x
.
Use trig identities.
How do we find
By integration.
Set that up.
Do the integration.
How do we find
?
We substitute our result for
into the equation for
We had
. Put this into
Do the substitution.
How do we find
?
By integration.
Set that up.
Do the integral.
With u = sin
x
, this is of the form
We get
Combine the results for
to get
y
p
.
This is the
particular solution.
Combine this with the homogeneous solution to get the full solution.
This is the
full solution.
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