490 Ordinary Differential Equations
EXERCISE 10.8
- Let Y1 = y and Y2 = y' and write equation (5) as an equivalent system of
two first-order differential equations.
For each of the following cases, display a graph of Y1 (t) = y(t) on [O, 10]
and display a phase-plane graph of Y2 versus Y1.
a. Let P = 0 (a linear spring) and K = A = w = 1. Use SOLVESYS
or your computer software to solve the system on the interval [O, 10] for
the initia l conditions y 1 (0) = 1 and y 2 (0) = 0 with damping constants
(i) C = 0 (no damping) and (ii) C = .5.b. Let P = .1 (a hard nonlinear spring) and K =A= w = 1. Numerica lly
solve the system on the interval [O, 10] for the initial conditions Y1 (0) = 1
and y 2 (0) = 0 with damping constants (i) C = 0 and (ii) C = .5.c. Let P = -.1 (a soft nonlinear spring) and K = A = w = 1. Solve
the system on the interval [O, 10] for the initial conditions Y1 (0) = 1 and
y 2 (0) = 0 with damping constants (i) C = 0 and (ii) C = .5.10.9 Van Der Pol's Equation
In 1921, E. V. Appleton and B. van der Pol initiated research on the oscil-
lations produced by electrical circuits which contain triode generators. Their
research lead to the study of the following nonlinear differential equation, now
known as van der Pol's equation
(1) x" + c(x^2 - l)x' + x = 0.
During 1926-27, van der Pol developed methods for solving equation (1). The
original electrical circuits studied by Appleton and van der Pol in the 1920 s
contained vacuum tubes. Today circuits which produce similar oscillations
occur on semiconductor devices. Van der Pol's equation also arises quite
often in nonlinear mechanics.
Letting Y1 = x and Y2 = x', we can rewrite equation (1) as the following
equiva lent system of two first-order equations
(2)