A simple solution procedure based on a Kerr-type differential relation is developed for axisymmetric indentation of an elastic thin layer on a rigid substrate. A key merit of our method over existing methods is that a simple Kerr-type differential relation between contact pressure and surface deflection of elastic layer holds both inside and outside the contact zone. With the aid of Betti's reciprocal theorem, explicit relations between indentation force, indentation depth and contact radius are derived for both compressible and incompressible elastic layers bonded or sliding on the rigid substrate. In particular, beyond the existing leading-order formulas, several higher-order formulas are derived which reduce to the existing leading-order formulas when higher-order terms are neglected. Compared to the existing leading-order formulas, our explicit higher-order formulas are in better agreement with accurate numerical results. The validity and accuracy of explicit formulas given by the present method are well verified by detailed comparison with available experimental data and accurate numerical results