200 Ordinary Differential Equations
- About 1600, the French lawyer, politician, and part-time mathemati-
cian, Frarn;ois Viete introduced a numerical method similar to, but more
cumbersome than, Newton's method for approximating roots of polyno-
mials. Perhaps Viete's method inspired Newton to invent his method of
1669. Find the roots of the following equations considered by Viete.
x^5 - 5x^3 + 500x = 7905504
x^6 - 15 x^4 + 85x^3 - 225x^2 + 274x = 120
- Evidently, the only equation which Newton, himself, ever solved using
Newton's method was x^3 - 2x - 5 = 0. Newton solved this problem in
- Of course, every numerical technique devised since that time has
been used to so lve this equation. Solve this cubic equation. - Find the roots of the quintic polynomial
p(x) = x^5 -(13.999+5i)x^4 +(7 4.99+55.998i)x^3 -(159.959+260.982i)x^2 +
(1.95 + 463.934i)x + (150 - 199.95i).4.3 Homogeneous Linear Equations with
Constant Coefficients
Many practical and important physical phenomena can be modelled by n-
th order linear differential equations with constant coefficients. We will now
show how a simple pendulum can be modelled by a second order homogeneo us
linear differential equation with constant coefficients.
Simple Pendulum A simple pendulum consists of a rigid straight wire
of negligible mass and length L with a bob of mass m attached to one end.
The other end of the wire is attached to a fixed support. The pendulum is
free to move in a vertical plane. Let B be the angle the wire makes with
the vertical- the equilibrium position of the system. We will choose B to be
positive if the wire is to the right of vertical and negative if the wire is to the
left of vertical. See Figure 4.4. If we neglect resistance due to friction and the
medium (usually air) in which the system is operating, then there are only
two forces acting on the mass m: Fw, the tension in the wire, which acts
along the wire and toward the support; and F 9 = mg, the force of gravity,
which acts vertically downward. The force of gravity may be represented as
two forces: one that acts parallel to Fw but in the opposite direction, FN;
and one that acts perpendicular to Fw, Fr. From Figure 4.4 we see that
FN = mgcosB = -Fw and Fr= mgsinB.