Computer Algebra Recipes

26 CHAPTER 1. PHASE-PLANE PORTRAITS
asymptotically approaches a stable focal point at the origin. This behavior is
characteristic of underdamping.
v
1
0.5
–1 –0.5 0 0.5 x 1
–0.5
–1
Figure 1.8: Phase-plane portrait for ¯ =3.
Finally, for ¯ = 3, the phase-plane trajectory shown in Figure 1.8 approaches a
stable nodal point at the origin, a behavior characteristic of overdamping. Itis
left as an exercise for you to determine the critical damping threshold between
under- and overdamping.
Unlike the phaseportrait command, DEplot mayalsobeusedtoproduce
solution curves, but not tangent ¯elds, for single higher-order ODEs.
Taking, say, ¯ = b2 =0:2, Vectoria now uses DEplot to generate the x(t)
solution curve shown in Figure 1.9 for the second-order ODE de, giventhe
initial condition 3 x(0) = 1, _x(0)=0. Thetimerangeisfromt =0to50,and
the time step size in the underlying numerical scheme is taken to be 0:05.
> beta:=b[2]:
> DEplot(de,x(t),t=0..50,[[x(0)=1,D(x)(0)=0]],stepsize=0.05);
The numerically derived solution curve in Figure 1.9 decreases in amplitude in
an oscillatory manner, again characteristic of the underdamped SHO.
Finally, since the SHO equation de is linear with constant coe±cients, an
analytic solution can be easily obtained for x(t) usingthedsolve command.
> dsolve(fde,x(0)=1,D(x)(0)=0g,x(t));
x (t) = 1 p
11 e
(¡
33
t
10 ) sin
Ã
3 p 11 t
10
!
t
+ e
(¡ 10 ) cos
Ã
3 p !
11 t
10
3 Note that the derivative condition _x(0) = 0 is entered as D(x)(0)=0, where D is the
di®erential operator. The di®erential operator D is more general than diff. It can represent
derivatives evaluated at a point and can di®erentiate procedures.