Constructing General Solution for Abel Second Type ODE from Particular Solutions

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SUMMARY

The discussion centers on constructing the general solution for an Abel second type ordinary differential equation (ODE) using particular solutions. It is established that, similar to the Riccati equation, it is feasible to derive the general solution from one or two particular solutions through integration techniques. The user references a manual that outlines this method, indicating that while three particular solutions are typically required for Riccati equations, fewer can suffice for Abel equations.

PREREQUISITES
  • Understanding of Abel second type ODEs
  • Familiarity with Riccati equations
  • Knowledge of integration techniques in differential equations
  • Experience with ODE solution manuals
NEXT STEPS
  • Research integration methods for deriving general solutions from particular solutions in ODEs
  • Study the properties and solutions of Riccati equations
  • Explore advanced techniques in solving Abel second type ODEs
  • Review relevant ODE solution manuals for practical examples
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Mathematicians, students of differential equations, and researchers focusing on ODE solutions, particularly those interested in Abel and Riccati equations.

smallphi
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I have Abel equation of second type. I've found one particular solution for sure, maybe two. Is it possible to construct the general solution from those with integration or whatever?

I know that is possible for Ricatti ODE and since Ricatti is particular case of Abel, I was wondering if similar tricks worked for Abel?
 
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It's possible with only two and even with only one, but it involves integration. I saw it in some manual for solving ODE's.
 
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