We consider an isotropic cylindrical shell of varying thickness and density along the generatrix. Let the shell be under pressure, which is symmetric and also varying along the generatrix. We follow the polupostamenty theory by V. Z. Vlasov and consider the problem of the dynamical stability of the shell. We obtain the exact solution corresponding to the certain relation between thickness, pressure and density. Such kind of shells of extent medium is important in mechanical and aerospace engineering for optimal mass obtaining. In the paper we obtain minimum values of the excitation coefficients for five boundary value problems, which are of great importance in engineering. We give the accuracy estimation of the WKB method for these problems. Numerical results are summarized in the table.