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Does anyone know of any nice techniques for solving differential equations of the form p(x)x'' = k for x(t)?

Get out the old slide rule

Look up a numerical methods text for solving nonlinear differential equations, I've forgotten most of this unfortunately.

x'' = k/p(x)

integrate both sides wrt x

integral( d(x')/dt , dx) = k integral(1/p(x) , dx)

hand wave the math to get

integral( (dx/dt) , d(x')) = k integral(1/p(x) , dx)
integral( x' , d(x')) = k integral(1/p(x) , dx)
(x'^2)/2 + C = k integral(1/p(x) , dx)

Solve 1st order diff eq, etc.

This is basically how one gets conservation of energy, etc

hand wave the math to get

lolwut

that doesn't seem right. proof plz

Bump for >>5

slightly more rigorously
integral( d(x')/dt , dx)
integral( d(x')/dt , x' dt)
integral( x' d(x')/dt , dt)
x' * x' - integral(d(x')/dt  x', dt) + C (integrate by parts)

equating last 2 lines,
2integral( x' d(x')/dt , dt) = x'^2 + C
integral( x' d(x')/dt , dt) = (x'^2)/2 + C
as claimed, with integration constant since all integrals are indefinite

RHS doesn't change