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Exercise 4.10: Electrical circuits.

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.