The general solutions to hyperbolic equations of the fourth and sixth orders are obtained using Vekua's method for the representation of the general solutions to elliptic equations of order $2n$ with the aid of $n$ analytic functions. It is assumed that the right-hand sides of the hyperbolic equations can be expanded in time series of sines. The systems of equations of various approximations for a prismatic thin body in terms of moments with respect to a system of Legendre polynomials can be reduced to these equations and to the hyperbolic-type equations of higher order.