This fourth volume in the series "Mathematics and Physics Applied to Science and Technology": (i) follows the first three volumes on complex, transcendental and generalized functions that provide a detailed background on the theory of functions; (ii) starts the next set "On Boundary and Initial Value Problems" with the "Ordinary Differential Equations with Applications to Trajectories and Oscillations". The book "Classification of Differential Equations and Examples of their Applications" is the sixth and last of volume IV, corresponding to the ninth book of the series. It consists of the final chapter 10 of the volume IV containing 20 example related to the preceding five books and chapters 1 to 9. It includes two recollections the first with a classification of differential equations into 500 standards and the second with a list of 500 applications.The ordinary differential equations are classified in 500 standards concerning methods of solution and related properties, including: (i) linear differential equations with constant or homogeneous coefficients and finite difference equations; (ii) linear and non-linear single differential equations and simultaneous systems; (iii) existence, unicity and other properties; (iv) derivation of general, particular, special, analytic, regular, irregular and normal integrals; (v) linear differential equations with variable coefficients including known and new special functions.The theory of differential equations is applied to the detailed solution of 500 physical and engineering problems including: (i) one- and multidimensional oscillators, with damping or amplification, with non-resonant or resonant forcing; (ii) single, non-linear and parametric resonance; (iii) bifurcations and chaotic dynamical systems; (iv) longitudinal and transversal deformations and buckling of bars, beams and plates; (v) trajectories of particles; (vi) oscillations and waves in non-uniform media, ducts and wave guides.