Solving the Equation y'(x) = 1/(x^N+1)

  • Context: Undergrad 
  • Thread starter Thread starter Gregg
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SUMMARY

The discussion focuses on solving the differential equation y'(x) = 1/(x^N + 1) using polynomial factorization and integration techniques. The participants detail the factorization of x^N + 1 into complex roots and demonstrate how to apply partial fraction decomposition to derive the integral. Specifically, they highlight the closed-form solution involving the hypergeometric function Hypergeometric2F1 and discuss the implications of integer values of N on the complexity of the solution.

PREREQUISITES
  • Understanding of complex numbers and their roots
  • Familiarity with polynomial factorization techniques
  • Knowledge of partial fraction decomposition
  • Basic concepts of hypergeometric functions, specifically Hypergeometric2F1
NEXT STEPS
  • Study the properties and applications of Hypergeometric2F1 functions
  • Learn about advanced integration techniques involving complex variables
  • Explore the implications of polynomial factorization in differential equations
  • Investigate the role of series expansions in solving integrals of rational functions
USEFUL FOR

Mathematicians, students of calculus and differential equations, and anyone interested in advanced integration techniques and polynomial analysis.

  • #31


Hello Gregg and Jackmell

Is that true for the general case? If so, sorry if I said it wasn't.

Yes that's true for the general case. But with + instead of - at denominator, the general term of the series will not be real. On theoretical viewpoint, it doesn't matter : after integration the general term will be a complex logarithm. On the other hand, further simplifications will be laborious.
In order to remain in the real range, it's better to start with a similar series, but with a quadric at denominator. The integation isn't too difficult and leads to a general term including real logarithm and arctangent. (i.e.: attachement)
 

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