Computer Algebra Recipes

4.2. SECOND-ORDER MODELS 173
An explicit solution is obtained by solving sol for x.
> x:=solve(sol,x);
x := ¡ 1 88 %1 + 8 %1
3
p 11 p 3 ¡ 77 ¡ 3 p 33
32 %1 ¡ 43 ¡ 5 p 33
%1 := JacobiSN
Ã
t
p
30 + 2 p 33
;
12
s
15 ¡ p 33
15 + p 33
The formidable-looking answer, which is challenging to derive with pen and
paper,isgivenintermsoftheJacobian elliptic sine function, whichMaple
expresses in the form JacobiSN(u; k). When written by hand, the elliptic sine
function is often written as sn(u), the argument k not being explicitly expressed.
How is sn(u) de¯ned? The upper limit Á in the ¯rst integral of equation
(4.11) is referred to as the \amplitude" of u and is written as Á =am(u).
Then, for a given k value, the elliptic sine function, sn(u), is de¯ned by
sn(u) =sinÁ = sin(am(u)): (4.12)
For k =0,itiseasytoverifythatsn(u) reduces to sin(u). As k is increased,
the elliptic sine function deviates away from the \ordinary" sine function shape.
To learn more about the Jacobian elliptic sine function, Mike suggests that you
try some of the relevant problems that follow this recipe.
The solution is now plotted over the time interval t = 0 to 30, displaying
asymmetric oscillation about the equilibrium point.
> plot(x,t=0..30,thickness=2);
0.2
0
–0.2
–0.4
!2
5 10 15 t 20 25 30
Figure 4.7: Asymmetric oscillations of the eardrum.
Unfortunately, Mike has to leave to pick Vectoria up, so let's wish him luck
in his quest for this fair damsel's hand and hope that he will return soon to
explore some more interesting recipes with us.