General Contents
Detailed Contents
Index
Programmed tutorial: Laplace Transform and Initial Value Problem 2
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(Based on a
similar problem
in
Paul’s Online Math Notes
.)
Use
Laplace
transforms to solve
General Contents
Detailed Contents
Index
What do we do first?
We take the
Laplace
transform of every term.
Set that up.
3
ℒ
{
y
′′} + 11
ℒ
{
y
′} – 4
ℒ
{
y
} =
ℒ
Which entries in the
Table of Laplace Transforms
apply here?
On the left-hand side of the equation, we can use #35, #36.
On the right-hand side of the equation, we can use #23.
Apply these entries.
We get
Substitute the initial values.
We get
Simplify and collect like terms.
The equation becomes
Solve for
Y
(
s
).
Simplify the numerator.
Distribute the multiplication in the numerator and factor the quadratic in the denominator.
Set up a
partial fraction
expansion here.
You may want to solve for
A
,
B
,
C
, and
D
yourself for
review
.
Otherwise, use this
solver
, with this input:
(-3s^2-24s-47) / (
(3s-1)
(s+4)^3
)
The result is
Have we solved the problem?
No.
What do we have at this point?
We have found the
Laplace
transform of the solution.
Which entries in the
Table of Laplace Transforms
apply here?
#2 and
#23.
Determine the solution of the original problem by finding the inverse
Laplace
transforms.
ℒ
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