In this series, I’m discussing how ideas from calculus and precalculus (with a touch of differential equations) can predict the precession in Mercury’s orbit and thus confirm Einstein’s theory of general relativity. The origins of this series came from a class project that I assigned to my Differential Equations students maybe 20 years ago.
We have shown that the motion of a planet around the Sun, expressed in polar coordinates with the Sun at the origin, under general relativity follows the initial-value problem
,
,
,
where , , , is the gravitational constant of the universe, is the mass of the planet, is the mass of the Sun, is the constant angular momentum of the planet, is the speed of light, and is the smallest distance of the planet from the Sun during its orbit (i.e., at perihelion).
In the two previous posts, we derived the method of undetermined coefficients for the simplified differential equations
.
and
.
In this post, we consider the simplified differential equation if the right-hand side has only the fifth term,
.
Let . Then satisfies the new differential equation . Also, . Substituting, we find
The characteristic equation of this new differential equation is
Therefore, the general solution of the new differential equation is
.
The constants and can be found by substituting back into the original differential equation:
Matching coefficients, we see that and . Therefore, the solution of the simplified differential equation is
.
In particular, setting and , we see that
is a particular solution to the simplified differential equation.
In the next post, we put together the solutions of these three simplified differential equations to solve the original differential equation,
.