We're required to analyse a particle moving in the potential U(x) = a/x^2 (a > 0). Setting F = U(x) and using the Newton equation F = ma, this gives rise to the DE:
d2x = a/m * (1/x^2)
I can't for the life of me figure out what method to use or even what sort of DE this is! That 1/x^2 is blowing my mind.
Any clues would really help.
The Attempt at a Solution
I've tried treating this as a non-homogeneous constant-coefficient ODE of the form
ax'' + bx' + cx = f(t)
(setting f(t) = 1/ x(t)^2 )
I try to solve
d2x = x(t)^-2
Ignoring constants for now.
Complementary solution (solution to the complementary equation:
ax"(t) + bx'(t) + cx(t) + d = 0)
-> x_c = at + b
-> xp = x(t)^-2 + x(t)^-1 + x(t) + c1 ????
Next I find the derivatives of the particular solution
x'p = -2x(t)^-3 * x'(t) -x(t)^-2 * x'(t) + x'(t)
And using the chain rule:
x''p = [ -2x(t)^-3 * x''(t) + x'(t) * 6 x(t)^-3 * x'(t)] + [-x(t)^-2 * x''(t) + x'(t)* 2x(t)^-3 * x'(t)] + x''(t)
At this point (right before finding constants) I seriously begin to doubt whether this is the method to use. Any hints?
I also tried using the modified Newton equation (t - t0 = +/- sqrt(m/2) int[1/sqrt(E-U(y)] dy... it's called 'reduction to quadrature') which yielded a very hard integral I couldn't do by substitution or parts
int ((E - a/y^2)^-1/2 ) dy
I'm yet to try partial fractions, but I seriously doubt whether that would work...