Question

Solve the PDE dz/dx+dz/dy = z^2

Answer 1

Solve the PDE dz/dx+dz/dy = z^2

Solution:

A parametrization invariant is:

$dx=dy=\frac{dz}{z^2}$

From $dx = du$ ,

$x-y=C_1$ which is a first characteristic equation.

From $dy=\frac{dz}{z^2}$ ,

$y=-\frac{1}{z}+C_2$ is the second characteristic equation.

The general solution of the PDE expressed on the form of an implicit equation is :

$\Phi((x-y),(y+\frac1z))=0$

where $\Phi$ is any differentiable function of two variables.

Answer: $\Phi((x-y),(y+\frac1z))=0$ .