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 \( g \) are arbitrary continuously differentiable functions. For this system, construct solutions that satisfy the conditions: \[ u(x, x)=\varphi(x), v(x,-x)=\psi(x), x \geq 0 \text {. } \]