11.6.5 Systems of Equations in Partial Derivatives of the First Order

Condition: Show that the general solution of the system \ (\ frac {\ partial u} {\ partial x}-\ frac {\ partial v} {\ partial y} = 0, \ quad \ frac {\ Partial U} {\ Partial y}-\ frac {\ partial v} {\ partial x} = 0 \), has the form \ (u (x, y) = f (x+y)+g (x-y), \ (v (x, y) = f (x+y) -g (x-y) \), where \ (f \) and \ (g (g (g (g. \)-arbitrary continuously differentiated functions. For this system, build solutions that satisfy the conditions: \ [u (x, x) = \ varphi (x), v (x, -x) = \ psi (x), x \ gq 0 \ text {. } \]