HP 40gs User Manual page 248

hp40g+.book Page 66 Friday, December 9, 2005 1:03 AM
14-66
ILAP is the inverse Laplace transform of a given
expression. Again, the expression is the value of a
function of the variable stored in VX.
Laplace transform (LAP) and inverse Laplace transform
(ILAP) are useful in solving linear differential equations
with constant coefficients, for example:
⋅ q y ⋅ f x ( )
y" + p y' + =
y 0 ( ) y' 0 ( )
= a =
The following relations hold:
+ ∞ x – t ⋅
∫
LAP(y)(x) = e
0
1
⋅
------- -
ILAP(f)(x) =
2iπ
where c is a closed contour enclosing the poles of f.
The following property is used:
( ) x ( ) y 0 ( )
LAP y' = –
The solution, y, of:
⋅ q y ⋅
y" + p y' + =
is then:
LAP f x ( ) ( ) (
+
⎛
------------------------------------------------------------------ -
ILAP
⎝
2
x + px
Example
To solve:
⋅ 9 y ⋅
y" 6 – y' + =
c
type:
LAP(X · EXP(3 · X))
The result is:
1
------------------------- -
2
x – 6x + 9
b
y t ( ) t d
zx
∫
f z ( ) z d
e
c
x LAP y ( ) x ( ) ⋅
+
f x ( ), y 0 ( ) a, y' 0 ( )
=
) a ⋅
x + p + b
⎞
⎠
+ q
3 x
⋅ , y 0 ( ) a, y' 0 ( )
x e =
Computer Algebra System (CAS)
= b
= b