The Laplace Trans! arm Method 239require using the linearity property or translation property of Laplace trans-
forms. To manually calculate the inverse Laplace transform often requires
using partial fraction expansion and the use of a table of Laplace transforms.!Comments on Computer Software I Algorithms for calculating the Laplace
transform and the inverse Laplace transform a re often included in computer
a lgebra systems (CAS). What the user needs to know in order to use such
a CAS is the command to use to invoke the Laplace transform or inverse
Laplace transform, the required arguments of the command, and the syntax
for entering the function f(x ) or F(s). A CAS will not show the computa-
tions used to a rrive at the answer, it will just provide the answer. Thus, if
a user specifies the proper syntax to request a CAS to calculate the Laplacetransform of f(x) = x^2 sinx the CAS will simply respond
2-1+3s^2
(1 + s2)3.
And if the user specifies the proper syntax to request a CAS to calculate the
2inverse La place transform of F( s) = ( ) , the CAS will respond 2 -2e-x.
ss+l
EXERCISES 5.1
In exercises 1-6 manually calculate the Laplace transform from its
definition. If you have a CAS available which calculates the Laplace
transform, also use the CAS to calculate the Laplace transforms of
exercises 1-6 and compare those answers to the ones you obtained
by hand.
- Let a and b be real constants.
a. Calculate .C [ x sin bx].
b. Use the translation property to calculate .C[xeax sin bx ].- Let a and b be real constants.
a. Calculate .C [ x cos bx].
b. Use the translation property to calculate .C[xeax cos bx].- Find .C[f(x)] for
{0,f(x ) =
1,0:::; x < 3
- Find .C[g(x)] for
{1 -x
g(x) = '
x - 1,0 :::; x < 1
1:::; x