A novel differential formulation of electromagnetic scattering by rotationally symmetric penetrable bodies is presented. The formulation is essentially based on the representation of vector fields in terms of spherical vector wave functions. In regions where we have physical inhomogenity the representation is in terms of position-dependent expansion coefficients. The differential equation system for these variable coefficients is deduced from the relevant wave equation. This system is numerically solved as an initial value problem, and the unknown constant expansion coefficients are found with a matrix inversion. The method was tested and validated on a number of problems, whose exact solutions are available, and the results obtained for scatterers for which no exact solutions are present were succesfully compared with the ones obtained by other methods.