General Contents
Detailed Contents
Index
Linear Nonhomogeneous Differential Equations and Undetermined Coefficients: Example 2
If you find this page helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
If you want to see all of the following steps at once, click the "All Steps" button. Otherwise, use the "Next" button.
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
.
We 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 Undetermined Coefficients
.
What is the form of the particular solution here?
We need a polynomial having the same degree as the term(s) on the right-hand side of the original differential equation.
Set that up.
How do we determine the coefficients
A, B, C, D,
and
E
?
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 the coefficients?
We can equate the coefficients of like terms.
Equate the coefficients of the “
x
n
” terms.
Solve for
A, B,
and
C.
We get
Combine these results for the coefficients to get
y
p.
This is the
Particular Solution.
Combine
y
h
and
y
p
to get the solution of the differential equation.
We get
How can we check this result?
We can find
and substitute in the original differential equation.
Determine
Find
Substitute these derivatives in the original differential equation.
We get
Collect like terms.
Does it check?
Yes, it equals the right-hand side of the original equation.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
.