General Solution for 2nd Order PDE: Is it Possible?

In summary, the conversation is about finding a general formulation for a PDE of order 2 and using Maple to calculate the solution. However, the solution is complex and requires a suitable transformation to one of the canonical representations (parabolic, hyperbolic, or elliptic) in order to be solved. This transformation involves eliminating the mixed and linear terms, resulting in an equation of the form ## Au_{xx}(x,y)+Cu_{yy}(x,y)+Fu(x,y)=g(x,y) ##.
  • #1
Jhenrique
685
4
Hellow everybody!

A simple question: exist a general formulation, a solution general, for a PDE of order 2 like:
## au_{xx}(x,y)+2bu_{xy}(x,y)+cu_{yy}(x,y)+du_x(x,y)+eu_y(x,y)+fu(x,y)=g(x,y) ##
?

The maple is able to calculate the solution, however, is a *monstrous* solution!
 
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  • #2
With a suitable transformation you can change it to one of the canonical representations of pdes (parabolic,hyperbolic and elliptic) and solve it.
 
  • #3
MathematicalPhysicist said:
With a suitable transformation you can change it to one of the canonical representations of pdes (parabolic,hyperbolic and elliptic) and solve it.

This transformation consists in eliminate the mixed and linear terms? Resulting an equation of kind:

## Au_{xx}(x,y)+Cu_{yy}(x,y)+Fu(x,y)=g(x,y) ##

?
 

1. Is there a general solution for 2nd order partial differential equations (PDE)?

Yes, there is a general solution for 2nd order PDEs. However, it may not always be possible to find an explicit solution, and in some cases, only an implicit solution can be obtained.

2. What is the process for finding the general solution of a 2nd order PDE?

The process for finding the general solution of a 2nd order PDE involves separating the equation into two ordinary differential equations, solving each one separately, and then combining the solutions to obtain the general solution.

3. Can a 2nd order PDE have multiple general solutions?

Yes, a 2nd order PDE can have multiple general solutions. This is because the general solution is a family of solutions that satisfies the given PDE, and there can be multiple ways to satisfy the equation.

4. Are all 2nd order PDEs solvable with the same method?

No, not all 2nd order PDEs can be solved with the same method. The method used to solve a 2nd order PDE depends on the specific form of the equation and the boundary conditions.

5. Can the general solution of a 2nd order PDE be expressed in terms of elementary functions?

In some cases, the general solution of a 2nd order PDE can be expressed in terms of elementary functions such as polynomials, logarithms, and trigonometric functions. However, in many cases, the general solution may involve special functions or cannot be expressed in terms of elementary functions at all.

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