Solving a PDE: Simplifying Third Order Equation to Second Order Airy Equation

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SUMMARY

The discussion centers on simplifying a third-order partial differential equation (PDE) into a second-order Airy equation. The original equation is given as \(\frac{\partial^{3}F(x,y)}{\partial y^{3}}-ix(y+C)\frac{\partial F(x,y)}{\partial y}=0\), where \(C\) is a positive constant and \(i=\sqrt{-1}\). The key suggestion is to define \(G(x,y) = \partial F(x,y)/\partial y\), transforming the problem into a second-order ordinary differential equation. The goal is to manipulate the term \(-ix(y+C)\) to match the standard form of the Airy equation, \(f(y)''-y*f(y)=0\).

PREREQUISITES
  • Understanding of partial differential equations (PDEs)
  • Familiarity with ordinary differential equations (ODEs)
  • Knowledge of Airy equations and their standard forms
  • Basic complex number theory, particularly the use of \(i=\sqrt{-1}\)
NEXT STEPS
  • Study the transformation techniques for PDEs to ODEs
  • Learn about the properties and solutions of Airy equations
  • Explore methods for simplifying complex expressions in differential equations
  • Investigate the role of boundary conditions in solving PDEs
USEFUL FOR

Mathematicians, physicists, and engineers dealing with differential equations, particularly those interested in the simplification and solution of PDEs and ODEs.

ssatonreb
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Hello!

I have difficulty to solve a PDE. I'm trying to simplify the third order eq. into the second order Airy equation. But I can't see where I could start. Could you, please, help me.
Where should I start?
Equations is:

[tex] \frac{\partial^{3}F(x,y)}{\partial y^{3}}-ix(y+C)\frac{\partial F(x,y)}{\partial y}=0[/tex]

where C is positive constant [tex]i=\sqrt{-1}[/tex]

Thank You.
 
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[tex]F(x,y)[/tex] itself does not appear in the equation. So for example define [tex]G(x,y) = \partial F(x,y)/\partial y[/tex] to get a second-order equation for [tex]G[/tex] . Also, there are no derivatives [tex]\partial/\partial x[/tex], so we might as well consider it an ordinary differential equation.
 
Thank You, but Airy equations is witten as f(y)''-y*f(y)=0.

How can I simplify eq. in order to -ix(y+C) become y?
 

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