Summary: A unique solvability of the Cauchy problem for a class of semilinear Sobolev-type equations of the second order is proved. The ideas and techniques, developed by G. A. Sviridyuk and T. G. Sukacheva [Differ. Equations 26, No. 2, 188–195 (1990); translation from Differ. Uravn. 26, No. 2, 250–258 (1990; Zbl 0707.34054)] for the investigation of the Cauchy problem for a class of semilinear Soboley-type equations of the first order and by A. A. Zamyshlyaeva and E. V. Bychkov [Vestn. Yuzhno-Ural. Gos. Univ., Ser. Mat. Model. Program. 18(277), No. 12, 13–19 (2012; Zbl 1413.35009)] for the investigation of the high-order linear Sobolev-type equations are used. We also use the theory of the differential manifolds which was finally formed in S. Leng’s work [Introduction to Differentialbe manifolds. New York: Springer -Verlag (2002)]. In the article, we consider two cases. The first one is when an operator at the highest time derivative is continuously invertible. In this case, for any point from a tangent bundle of an original Banach space, there exists a unique solution lying in this space as trajectory. The second case, when the operator is not continuously invertible, is of great interest for us. Hence we use the phase space method. Besides the Cauchy problem we consider the Showalter-Sidorov problem. The last generalizes the Cauchy problem and is more natural for Sobolev-type equations. In the last section we describe an algorithm for the numerical solution of the Showalter-Sidorov problem for Sobolev-type equations of the second order.