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Linear Nonhomogeneous Differential Equations and Undetermined Coefficients: Example 8
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Solve
.
What type of differential equation is this?
It is a linear nonhomogeneous differential equation with constant coefficients.
How do we solve it?
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?
We find the
characteristic equation
.
Set up that equation.
Solve for
m
.
m =
+2 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?
From the fifth row of the table in the above link,
How do we determine the coefficients
A
and
B
?
We find
and
, and substitute into the differential equation. Then by equating coefficients of like terms we can determine values for
A
and
B
.
Find
Factor the exponential and collect like terms.
We get
Find
Collect like terms.
We get
Susbstitute these results in the differential equation.
Combine like terms.
How can we use this result to find
A
and
B
?
We can equate the coefficients of like terms.
Equate the coefficients of the “
” terms.
We get
Solve for
A
.
Using this result, equate the constants from both sides.
We get
,or
Combine these results for
A
and
B
to get
.
This is the
Particular Solution
.
Combine
and
to get the solution of the differential equation.
The end.
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General Contents
Detailed Contents
Index
.