Computer Algebra Recipes

174 CHAPTER 4. NONLINEAR ODE MODELS
PROBLEMS:
Problem 4-20: Di®erent amplitude
Taking the initial conditions x(0) = ¡ 4
5 ,_x(0) = 0 in the text recipe, determine
the analytic form of the period and its numerical value. Determine the analytic
solution and plot it over a suitable time range.
Problem 4-21: Maximum energy for bounded motion
What is the maximum total energy E for which oscillatory motion of the
eardrum can occur in the text recipe? What are the numerical values of the
turning points?
Problem 4-22: Higher-order correction to the period
For the freely vibrating eardrum, also keep the cubic term in the Taylor expansion
of the force law and express the restoring force per unit mass as
F = ¡! 2 0 x ¡ ¯x2 ¡ °x3 . Taking ! =1,¯ = 3 1
1
4 , ° = 2 ,andA = 3 determine
the period of vibrations of the eardrum. By how much is the period
changed with respect to that predicted by the linear Hooke's law?
Problem 4-23: Plotting the elliptic sine function
Plot the Jacobian elliptic sine function sn(u) over the range u =0to20inthe
same graph for k =0:1, k =0:5, k =0:9, and k =0:995. Discuss the behavior
of sn(u) ask is increased.
Problem 4-24: Plotting the elliptic cosine function
In analogy to the elliptic sine function, the Jacobian elliptic cosine function of u
is de¯ned as cn(u) =cosÁ =cos(am(u)): Noting that the Maple command for
cn(u) isJacobiCN(u,k), plot the elliptic cosine function over the range u =0
to 20 in the same graph for k =0:1, k =0:5, k =0:9, and k =0:995. Discuss
the behavior of cn(u) ask is increased.
Problem 4-25: Properties of elliptic functions
Using the de¯nition of cn(u) in the previous problem and de¯ning another
elliptic function dn(u) = p 1 ¡ k2 sin 2 Á = p 1 ¡ k2 sn2 (u); prove the following:
(a) d
(cn(u)) = ¡ sn(u) dn(u);
du
(b) d2
du 2 (cn(u)) = (2 k2 ¡ 1) cn(u) ¡ 2 k 2 cn 3 (u);
Z
(c) cn(u) du = 1
k arccos(dn(u)).
Problem 4-26: Vibrating hard spring
The equation of motion for a \hard" spring is
Äx(t)+(1+a 2 x(t) 2 ) x(t) =0:
Analytically determine the period and the solution x(t) for the hard spring,
given the initial conditions x(0) = A, _x(0) = 0. Plot the solution over several
cycles for a = A =1.