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If the acceleration of an object due to gravity is given to be some function proportional to the inverse square of the distance of the object from the primary mass does it then follow that one solution to the position function would some how be related to the function

-ln(r)

as integrating with respect to r twice would give you that as a solution?

If so how does one interpret what that means?

I am new to differential equations and mechanics in general but I would appreciate it if someone could help me expand this idea and where I am going wrong.

Thanks!