488 Ordinary Differential Equations
for (a) A -.5 and (b) A = -1 for the initial conditions Y1(0) = 0 and
(i) y 2 (0) = 2.5 and (ii) y2(0) = 3.5. Display a phase-plane graph of Y2 versus
Y1 for each solution.
- Solve system (8), the simple pendulum with constant forcing function
model, on the interval [O, 10] when C = .5 for (a) A= -.5 and (b) A= -1
for the initial conditions y 1 (0) = 0 and (i) y2(0) = 2.5 and (ii) Y2(0) = 3.5.
Display a phase-plane graph of Y2 versus Y1 for each solution. - Use computer software to solve system (10), the variable length pendulum
model, on the interval [O, 10] with no damping (c = 0) and no forcing func-
tion (f(t) = 0) for the initial conditions y 1 (0) = 0, y2(0) = 1.5 for f(t) =
g(l + .1 sin wt) where (i) w = .5, (ii) w = .75, and (iii) w = .9. In each case,
display a graph of y 1 (t). Is the solution periodic? Display a phase-plane graph
of Y2 versus Y1.
- Numerically solve system (10), the variable length pendulum model, on the
interval [O, 10] with no forcing function (f(t) = 0), with damping coefficient
c =. lm, and with f( t) = g(l + .1 sin t) for the initial conditions y 1 (0) = 0 and
(i) y2(0) = 1.5 and (ii) y2(0) = 2.5. Display graphs of Y1(t) and phase-plane
graphs of Y2 versus Y1. - Solve system (10), the variable length pendulum model, on the interval
[O, 10] for the initial conditions y 1 (0) = 0 and Y2(0) = 2.5 when f(t) =
g(l + .lsint), when (i) c = 0 and (ii) c = .lm, and f(t) = mgsin{3t where
(a) {3 =. 5 and (b) {3 = l. Display graphs of y 1 ( t) and phase-plane graphs of
Y2 versus Y1.
- Let Y1 = x, Y2 = x', y3 = y, and y4 = y'. Write system (11), the Foucault
pendulum model, as a system of four first-order differential equations. Nu-
merically solve the resulting system on the interval [O, 10] for initial conditions
Y1(0) = 1, Y2(0) = 0, Y3(0) = 0, and y4(0) = 0 if f = 9.8 m and (a)</>= 0°,
(b) </> = 45°, (c) </> = -45°, (d) </> = 60°, and (e) </> = 90°. The period of a
pendulum with length f = 9.8 mis 21f, or approximately 6.28 seconds. From
the value of y 3 (6.28) estimate how long it will take the plane of swing of the
pendulum to appear to rotate 360° in cases (a)-(e).
- When the spring pendulum is at rest in the equilibrium position y 1
Lo+ mg/k and Y2 = y3 = y4 = 0. Solve system (13) on the interval [O, 10] for
Lo= 1.02 decimeters, g =. 98 decimeters/second^2 , and m/k = 1 second^2 for
the following initial conditions:
(i) Y1(0) = 2, Y2(0) = 0, y3(0) = .1, and y4(0) = 0.
(The spring pendulum is set into motion by pulling the mass to the right
so that B =. 1 radian and then releasing the mass without compressing
or elongating the spring.)