11.5.4.21 With constant coefficients

Problem: - Find the general formula for solving the problem with given data on the curve; - define and depict on the plane the area in which the classical solution exists and is unique for any initial data: - if the data is given on both branches of the hyperbola, - if the data is given on the branch of the hyperbola, indicated in the table, - if the data is specified on the fragment of this branch, specified in the table; - find the solution for the initial data, given in the table; - determine whether this solution continues beyond the found areas and is unique there. \[ \begin{array}{l} u_{x y}-u_{y y}=0 .\left\{\begin{array}{l} \left.u\right|_{\Gamma}=\varphi(x) \\ \left.u_{x}\right|_{\Gamma}=\psi(x) \end{array} \quad \Gamma: x(x+y)=-1,\right. \\ \left\{\begin{array}{ll} \varphi(x)=x-1 & \text { the lower branch } \\ \psi(x)=-1 & \text { the fragment } x \in[3,4] \end{array}\right. \\ \end{array} \]