HP F2226A - 48GII Graphic Calculator - Page 491

Page 16-13
L{df/dt} = s⋅F(s) - f
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
.
• 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 2 – As a follow up to Example 1, the acceleration a(t) is defined as
a(t) = d
2
r/dt
2
. If the initial velocity is v
o
= v(0) = dr/dt|
t=0
, then the Laplace
transform of the acceleration can be written as:
A(s) = L{a(t)} = L{d
2
r/dt
2
}= s
2
⋅R(s) - s⋅r
o
– v
o
.
• Differentiation theorem for the n-th derivative
.
Let f
(k)
o
= d
k
f/dx
k
|
t = 0
, and f
o
= f(0), then
L{d
n
f/dt
n
} = s
n
⋅F(s) – s
n-1
⋅f
o
−…– s⋅f
(n-2)
o
– f
(n-1)
o
.
• Linearity theorem
. L{af(t)+bg(t)} = a⋅L{f(t)} + b⋅L{g(t)}.
• Differentiation theorem for the image function
. Let F(s) = L{f(t)}, then
d
n
F/ds
n
= L{(-t)
n
⋅f(t)}.
Example 3 – Let f(t) = e
–at
, using the calculator with ‘EXP(-a*X)’ ` LAP, you
get ‘1/(X+a)’, or F(s) = 1/(s+a). The third derivative of this expression can
be calculated by using:
‘X’ ` ‚¿ ‘X’ `‚¿ ‘X’ ` ‚¿ µ
The result is
‘-6/(X^4+4*a*X^3+6*a^2*X^2+4*a^3*X+a^4)’, or
d
3
F/ds
3
= -6/(s
4
+4⋅a⋅s
3
+6⋅a
2
⋅s
2
+4⋅a
3
⋅s+a
4
).