Inhomogenous NON-linear differential equation

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SUMMARY

The discussion centers on solving an inhomogeneous non-linear differential equation of the form \(\ddot{x}+f(x)\dot{x}^2 = g(x)\). The equation resembles a Bernoulli equation, and the user suggests that a clever substitution, specifically \(x'' = x'(dx'/dx)\), may simplify the problem. Despite the user's belief in the existence of an exact solution, they have not yet succeeded in finding it. The conversation seeks insights on potential methods for solving this type of differential equation.

PREREQUISITES
  • Understanding of non-linear ordinary differential equations (ODEs)
  • Familiarity with Bernoulli differential equations
  • Knowledge of substitution methods in differential equations
  • Basic calculus and differential calculus concepts
NEXT STEPS
  • Research methods for solving non-linear ODEs
  • Explore advanced techniques for Bernoulli equations
  • Study the application of substitutions in differential equations
  • Investigate numerical methods for approximating solutions to complex ODEs
USEFUL FOR

Mathematicians, physicists, and engineering students dealing with non-linear dynamics and differential equations will benefit from this discussion.

Thoughtknot
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I'm having some trouble solving an equation that is similar to a Bernoulli equation. It is of the form

\begin{equation}
\ddot{x}+f(x)\dot{x}^2 = g(x)
\end{equation}

Where x is a function of time, perhaps. I feel moderately certain that there should exist an exact solution, but I've so far been unable to find it, and I have not run into any great amount of non-linear ODEs before.

Does anyone have any idea if it can be solved? Could it be solved by some clever substitution?
 
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The substitution x'' = x'(dx'/dx) would reduce this to a Bernoulli differential equation.
 

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