Abstract <p>The existence and uniqueness of the solution of the boundary value problem for the equation of mixed parabolic-hyperbolic type of the third order with variable coefficients at the lower terms is proved. By methods of reducing the order of the equation, the problem is reduced to the Cauchy problem for the equation of hyperbolic type and the first boundary value problem for the equation of parabolic type. The solvability of the problem is reduced to the solvability of the Fredholm integral equation of the second kind. After determining the trace of the function and its derivative with respect to y of the first order, the solution of the problem is completely determined in the areas under consideration.</p>