Is there a name for the DE given by Newton's law of gravity:
[tex]f''(x) f(x)^2 = -1[/tex]
Forget about the physics, this is a non linear second order DE. The given DE looks like it can be solve by separating the variables.
I don't know of any specific "name" for it but. since the independent variable, x, does not appear explicitely, it can be solved using "quadrature":
Let y= f'(x). Then f"(x)= y'= dydx= (dy/df)(df/dx) (by the chain rule)= (dy/df)y. Thus, your equation becomes yf^2 dy/df= -1, a separable first order equation. y dy= -df/f^2. Integrate both sides to find y as a function of f and then integrate y= f(x) to find f.
Separate names with a comma.