In modeling circuits containing devices whose electrical properties are current dependent, ODEs of the form

occur. For the case where

compute , . Plot , over the interval on separate graphs using ``xmgr.'' Use RKSUITE in conjunction with a linear system solver.

** 4.11** The orbit of the planet Mercury around the
Sun can be represented
as the solution to the differential equation,

where and **r** denotes the distance from the Sun
to Mercury. Here
is an angle in the plane of the orbit, is the gravitational
constant, **h** is the angular momentum, and is a parameter
determined by the effects of other planets on Mercury as well as the
Sun's oblateness, and a correction required by the general theory of
relativity. To solve this problem, we convert it to the first order system

where and .

To illustrate the phenomenon of * precession*,
choose ,
, , , and integrate the system over
several revolutions. Plot vs.
using ``xmgr''.
The plot should show that Mercury moves on an ellipse that is slowly rotating
in the orbital plane. The points of closest approach to the Sun are called
* perihelia*; the precession of these points is due to the perturbing
nonlinearity in the differential equation. The observed precession of the
perihelion of Mercury could not be explained by Newtonian mechanics and
remained a puzzle for many years. The closest agreement between observations
and the orbit modeled by the differential equation with containing
a relativistic correction is one of the major experimental confirmations of
Einstein's theory of general relativity.