MathDB

Given

x=\phi(u,v)

and

y=\psi(u,v)

, where

\phi

and

\psi

are solutions of the partial differential equation

(1) \;\,\;\, \; \frac{ \partial \phi}{\partial u} \frac{\partial \psi}{ \partial v} - \frac{ \partial \phi}{\partial v} \frac{\partial \psi}{ \partial u}=1.

By assuming that

x

and

y

are the independent variables, show that

(1)

may be transformed to

(2) \;\,\;\, \; \frac{ \partial y}{ \partial v} =\frac{ \partial u}{\partial x}.

Integrate

(2)

and show how this effects in general the solution of

(1)

. What other solutions does

(1)

possess?

Problem Statement