Name for DE given by law of gravitation

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SUMMARY

The differential equation (DE) derived from Newton's law of gravitation is expressed as f''(x) f(x)^2 = -1. This non-linear second-order DE can be solved by separating variables, despite the independent variable x not appearing explicitly. By substituting y = f'(x), the equation transforms into a separable first-order equation, allowing for integration to find y as a function of f, followed by integration to determine f(x).

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  • Understanding of differential equations, specifically non-linear second-order DEs.
  • Familiarity with the method of separation of variables.
  • Knowledge of integration techniques in calculus.
  • Basic concepts of Newtonian physics related to gravitation.
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  • Study the method of quadrature in solving differential equations.
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Mathematicians, physicists, and engineering students interested in solving differential equations, particularly those related to gravitational laws and their applications in various fields.

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Is there a name for the DE given by Newton's law of gravity:

f''(x) f(x)^2 = -1

?
 
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Tac-Tics said:
Is there a name for the DE given by Newton's law of gravity:

f''(x) f(x)^2 = -1

?

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.
 
Tac-Tics said:
Is there a name for the DE given by Newton's law of gravity:

f''(x) f(x)^2 = -1

?
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.
 

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