Solving Non-Linear Differential Equation with Fourier Transforms

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SUMMARY

The discussion focuses on solving the non-linear differential equation given by 1/2(f')^2 = f^3 + (c/2)f^2 + af + b using Fourier Transforms. The user highlights that while the equation is separable and can theoretically be solved through direct integration, the resulting integral involves complex elliptic functions, rendering practical solutions difficult. Specific values for the constants a, b, and c may simplify the equation, but general solutions remain challenging.

PREREQUISITES
  • Understanding of non-linear differential equations
  • Familiarity with Fourier Transforms
  • Knowledge of elliptic functions
  • Basic calculus and integration techniques
NEXT STEPS
  • Explore Fourier Transform techniques for differential equations
  • Research methods for solving elliptic integrals
  • Investigate specific cases of non-linear ODEs with known constants
  • Study the implications of boundary conditions on differential equations
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Mathematicians, physicists, and engineers dealing with non-linear differential equations, particularly those interested in advanced integration techniques and Fourier analysis.

Gengar
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Hiya. I have to solve this bad boy under the assumptions that f, f' and f'' tend to 0 as |x| tends to infinity:

1/2(f')^2 = f^3 + (c/2)f^2 + af + b

where a,b,c are constants. My thoughts are use Fourier Transforms to use the assumptions given, but not sure how to do them on these terms. Thanks!
 
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Hi !

This is a separable ODE which (in theory) can be solved by direct integration :
df / sqrt(2 f^3 + c f^2 + 2af + 2b) = dx
But the integral involves very complicated elliptic functions, so that it will be of no use in practice.
In particular cases, i.e. for some particular values of a, b, c, it might reduce to simpler functions.
 

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