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

[tex]f(x,y,z,p,q) = 0[/tex]

where [itex]p = \partial z/\partial x[/itex] and [itex]q = \partial z/\partial y[/itex], 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 [itex]z = F(x,y,a)[/itex] be a one parameter family of solutions of the above PDE, parametrized by [itex]a[/itex]. Then the envelope of this family, if it exists, also satisfies the PDE. What is the geometrical significance of this theorem?

Thanks.

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# Integrals of PDEs (help needed to interpret theorem)

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