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## Main Question or Discussion Point

I was thinking about a situation like this: You have a planet that has a nonuniform density (mass) like this [tex]\rho = a - r[/tex] where a is just some number and r is the radius away from the center of the planet. Then you have an object falling through the planet. I had no idea how to solve this differential equation. I don't even know if there is a solution. You'd have an F=ma like this: [tex]-\frac{G(a - r)(\frac{4}{3}\pi r^3)m}{r^2} = ma[/tex]. That becomes through simplification: [tex]\frac{d^2 r}{dt^2}=\frac{4}{3}\pi a G(r^2 - r)[/tex]. I don't really know much beyond solving simple differential equations. Is this solvable? If there is a solution, how do you get it?