What is the general solution for PDEs in the form of a question?

[Bruno Tolentino]

y = a x² + b x + c is a parabola. But, a parabola is just a kind of conic.

All conics are given by a x² + b x y + c y² + d x + e y + f = 0

The same way, the graphic y = f(x), with f(x) satisfying a d²f/dx² + b df/dx + c f = 0, is just a particular graphic of F(x,y) = 0 with F(x,y) satisfying

a d²F/dx² + b d²F/dxdy + c d²F/dy² + d dF/dx + e dF/dy + f F = 0

OBS: a, b, c... are constants.

So, which is the general solution for the PDE above? And, where I can visualize the graphic?

 

[HallsofIvy]

I am not sure I understand your question. It should be clear that the "general solution" to "[itex]a F_{xx}+ b F_{xy}+ c F_{yy}+ d F_x+ e F_y+ fF= 0[/itex]" is simply "[itex]ax^2+ bxy+ cy^2+ dx+ ey+ f= 0[/itex]".

 

[Bruno Tolentino]

I simply want to know which is the funcion F(x,y) that satisfies a PDE of second order with constant coeficients.

If this function F don't exist, or wasn't discoverd yet, so, exist some program that outline the graphic(approximately) of the solution of the PDE?

 

[HallsofIvy]

I simply want to know which is the funcion F(x,y) that satisfies a PDE of second order with constant coeficients.

If this function F don't exist, or wasn't discoverd yet, so, exist some program that outline the graphic(approximately) of the solution of the PDE?

That is basically the opposite of the question you originally asked! Second order partial differential equations can have very different solutions and are basically divided into "elliptic", "hyperbolic", and "parabolic" equations. Any introductory PDE textbook will have a discussion of what those type of equations and how their solutions differ.

 

[Bruno Tolentino]

Still today, none made some effort for try to unify these 3 kinds of solutions. Don't exist a general solution?

 

[micromass]

Still today, none made some effort for try to unify these 3 kinds of solutions. Don't exist a general solution?

No, it doesn't exist.

 

[Bruno Tolentino]

Why not? Is it impossible?

 

[micromass]

Yes, I think it is impossible to give a nice unified solution. But I have not seen this in a theorem/proof form yet. You can of course solve any PDE as in the original post if you are given specific values and specific boundary values. Just as conic section can have very different behavior, so do PDE's.