General Contents
Detailed Contents
Index
Linear Nonhomogeneous Differential Equations and Undetermined Coefficients: Example 4
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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.
How do we solve it?
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 try e
mx
in the differential equation and find the
characteristic equation
.
Set up that equation.
Solve for
m
.
Factor to get
So
m
= 0 or -2.
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 Undetermined Coefficients
.
What is the form of the particular solution here?
We need multiples of the cosine and sine functions with the same argument as on the right-hand side of the original differential equation.
Set that up.
How do we determine the coefficients
A
and
B
?
We find
y
p
and
y
p
, and substitute into the differential equation.
Find
y
p
Find
y
p
Substitute these results in the differential equation.
How can we use this result to find
A
and
B
?
We can equate the coefficients of like terms.
Equate the coefficients of the sin
x
terms.
Solve for
B
.
Equate the coefficients of the cos
x
terms.
We get
Combine these two results to solve for
A
and
B
.
We get
, or
Combine these results for
A
and
B
to get
y
p.
This is the
Particular Solution.
Combine
y
h
and
y
p
to get the solution of the differential equation.
How can we check this result?
We can find
Determine
Find
Substitute these derivatives in the original differential equation.
We get
Collect like terms.
We get
Does it check?
Yes, it matches the right-hand side of the original differential equation.
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.