Solutions of two plane elastic problems on dynamic pressure of a rigid punch on a half-space are described. Integral equations of mixed boundary value problems are obtained through the use of the Fourier and Carson transforms. The emphasis is put on the method of solving these equations. The solution is constructed as a series of transforms at small and large times, i.e., as two asymptotics. Numerical results presented by the authors show that a few first terms of these asymptotics give solutions which coalesce at a middle part of the time interval.