HP 50G, User Manual, Page: 489 / 887

Page 16-12
Example 3 – Determine the inverse Laplace transform of F(s) = sin(s). Use:
‘SIN(X)’
`
ILAP. The calculator takes a few seconds to return the result:
‘ILAP(SIN(X))’, meaning that there is no closed-form expression f(t), such that f(t)
= L
-1
{sin(s)}.
Example 4 – Determine the inverse Laplace transform of F(s) = 1/s
3
. Use:
‘1/X^3’
`
ILAP
μ
. The calculator returns the result: ‘X^2/2’, which is
interpreted as L
-1
{1/s
3
} = t
2
/2.
Example 5 – Determine the Laplace transform of the function f(t) = cos (a
⋅
t+b).
Use: ‘COS(a*X+b)’
`
LAP . The calculator returns the result:
Press
μ
to obtain –(a sin(b) – X cos(b))/(X
2
+a
2
). The transform is interpreted
as follows: L {cos(a
⋅
t+b)} = (s
⋅
cos b – a
⋅
sin b)/(s
2
+a
2
).
Laplace transform theorems
To help you determine the Laplace transform of functions you can use a number
of theorems, some of which are listed below. A few examples of the theorem
applications are also included.
Θ
Differentiation theorem for the first derivative. Let f
o
be the initial condition
for f(t), i.e., f(0) = f
o
, then
L{df/dt} = s
⋅
F(s) - f
o
.
Θ
Differentiation theorem for the second derivative. Let f
o
= f(0), and (df/dt)
o
= df/dt|
t=0
, then L{d
2
f/dt
2
} = s
2
⋅
F(s) - s
⋅
f
o
– (df/dt)
o
.
Example 1 – The velocity of a moving particle v(t) is defined as v(t) = dr/dt,
where r = r(t) is the position of the particle. Let r
o
= r(0), and R(s) =L{r(t)}, then,
the transform of the velocity can be written as V(s) = L{v(t)}=L{dr/dt}= s
⋅
R(s)-r
o
.