The relation between the boundary values of the Goursat problem with normal derivatives of the third order (Q1589114)

Let \(u(x,y)\) be a solution of the Goursat problem in the domain \(D=(0,x_1) \times(0,y_1)\) for the equation \(u_{xy}+au_x+ bu_y+cu=0\) with the boundary conditions \(u(x,0)= \psi(x)\), \(u(0,y)= \varphi(y)\), \(\varphi(0)= \psi(0)\), \(a,b\in C^3(\overline D)\), \(c\in C^2(\overline D)\), \(a^2+b^2+c^2\neq 0\). The relation between \(\psi(x)\), \(\varphi(y)\) and the normal derivatives \(\mu(x)= u_{yyy}(x,0)\), \(\lambda(y)= u_{xxx}(0,y)\) is established.