The unsteady laminar boundary layer on a deformable body in non uniform motion $U(x,t)=\Omega(t)V(x,t)$ is discussed. An universalisation of the unsteadyboundary layer equations is first made in the sense that neither equations nor boundary conditions depend on particular problem data The universality is achieved by tiansfering sets of parameters which express the influence of time and deformability conditions, charasteristic for each particular problem, into the independent variables. Subsequently, the solution of the universal equation is found in the form of series expansions in mentioned parameters.