Integrals of PDEs (help needed to interpret theorem)

[maverick280857]

Hello friends,

I'm reading about PDEs and my textbook lists 'integrals' of the pde

$$f(x,y,z,p,q) = 0$$

where $p = \partial z/\partial x$ and $q = \partial z/\partial y$, as

1. Complete Integral
2. General Integral
3. Singular Integral
4. Special Integral (solution that can't be classified into the above three categories...and can't be obtained from the general integral).

Specifically, I have the following questions:

1. What is the geometrical/physical significance of each of these 'integral' solutions, esp the singular solution of the PDE?

2. Let $z = F(x,y,a)$ be a one parameter family of solutions of the above PDE, parametrized by $a$. Then the envelope of this family, if it exists, also satisfies the PDE. What is the geometrical significance of this theorem?

Thanks.

 

[zhentil]

I wouldn't sweat the geometric significance of this. You're dealing with 5 dimensions. What helped for me was to get my geometric intuition from the quasi-linear case, and just know that the analysis works for the fully non-linear case.

 

[epenguin]

Part answer and new question

It is not 5-D it is 3-D surely?

But it would be good to get it straight for 2D firstly, then AFAIK there are not big differences of principle for other D but I stand to be corrected.

It would be pretty obvious to you if you found them illustrated, or if you use a graphic plotter and plot a family $F(x,y,a) = 0$
For 2D and equation f(x, y, dy/dx) = 0, with a family of solutions g(x, y, c) = 0 depending on a parameter a, you may have a continuous curve (called the envelope) whose slope at every point is equal to that in the parameter dependent solution. Therefore this curve also satisfies the differential equation f = 0. Better an example: for instance to the equation

dy/dx = y^(1/2)

there is the family of solutions y = (x – a)^2 . Each of these solutions is a curve with a minimum at y = 0, i.e. dy/dx = 0 at y = 0 so satisfying also the d.e. just given. y = 0 is a singular solution of the given d.e.


I always found singular solutions appealing because of their unlooked-for character. I wonder if there are any examples of where they are of any interest in general dynamics? A point traveling along a general solution and hopping onto a singular one when it mneets it? It should only be able to hop off if and when it gets back to the original point. Imagine there are no examples.

 

[zhentil]

For a general first-order equation in two variables, you have to solve a system of 5 ODEs.