Hello friends, 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.