A Cauchy-type problem for Euler-Poisson-Darboux equation (Q677752)

For Cauchy-type problems for the equation \[ u_{xx}- u_{yy}- {2q\over y} u_x- {2p\over y} u_y=0\quad\text{in }\{(x,y)\mid y<0\} \] solutions are given by explicit integral expressions involving the parameters \(\alpha:= p+q\), \(\beta:= p-q\). Conditions are formulated which guarantee that the given solutions satisfy the initial conditions in the classical sense.