Computer Algebra Recipes

3.2. SECOND-ORDER MODELS 121
3.2.2 Meet Mr. Laplace
The weight of evidence for an extraordinary claim must be
proportioned to its strangeness (known as the principle of Laplace).
Pierre-Simon Laplace, French mathematician (1749{1827)
Pierre-Simon Laplace was one of the most in°uential scientists in history. Indeed,
he has been referred to as the Newton of France. Not only did he make
outstanding contributions to astronomy, for example, mathematically investigating
the stability of the solar system, he developed probability theory as a
coherent body of knowledge from a set of miscellaneous problems, and played a
leading role in forming the modern discipline of mathematical physics. One of
his most important contributions to the latter is the Laplace transform, which
canbeusedtosolvelinearODEs.
The Laplace transform of a function f(t) isde¯nedas
L(f(t)) ´ F (s) =
Z 1
0
f(t)e ¡st dt:
By integrating by parts and assuming that e ¡stf(t) ! 0ast !1,itiseasy
to show that
³ ´
³ ´
L f(t) _ = sF(s) ¡ f(0) and L Äf(t) = s 2 F (s) ¡ sf(0) ¡ _ f(0):
The Laplace transform method of solving a linear ODE (system) with constant
coe±cients is to Laplace transform the ODE, solve the resulting algebraic
equation for F (s), and then perform the inverse Laplace transform to obtain
the solution f(t). In the era before computer algebra existed, tables of Laplace
transforms and their inverses were almost as common as integral tables, and
one of their main uses was in solving linear ODEs. With the Maple computer
algebra system, these tables are obsolete, since Maple has an integral transform
library package that includes the Laplace transform and its inverse.
The use of this package is now demonstrated in the ¯rst of the two recipes
that follow. It is desired to solve the following forced oscillator ODE for x(t):
Äx +4_x +13x =sin(t); with x(0) = 1 and _x(0) = ¡5:
Entering the integral transform package and using a semicolon to display
its contents, we see that it contains the Laplace transform (laplace) command
and its inverse (invlaplace).
> restart: with(inttrans);
[addtable; fourier; fouriercos; fouriersin; hankel; hilbert; invfourier;
invhilbert; invlaplace; invmellin; laplace; mellin; savetable]
The forced oscillator ode is entered,
> ode:=diff(x(t),t,t)+4*diff(x(t),t)+13*x(t)=sin(t);
μ μ
2 d d
ode := x (t) +4 x (t) +13x(t) =sin(t)
dt2 dt