Generalized solution of the initial-boundary-value problem for the wave equation with a mixed derivative and a general potential

We study the initial-boundary-value problem in a half-strip for a second-order inhomogeneous hyperbolic equation with constant coefficients and a nonzero potential containing a mixed derivative. The equation considered is the equation of transverse vibrations of a moving finite string. The problems with general initial conditions (nonzero string profile and nonzero initial velocity of string points) and fixed ends (Dirichlet conditions) are examined. Theorems on the existence and uniqueness of a solution are formulated and formulas for the solution are obtained.