The construction of mathematical models to address real-world problems has been one of the most important aspects of each of the branches of science. It is often the case that these mathematical models are formulated in terms of equations involving functions as well as their derivatives. Such equations are called differential equations. If only one independent variable is involved, often time, the equations are called ordinary differential equations. The course will demonstrate the usefulness of ordinary differential equations for modelling physical and other phenomena. Complementary mathematical approaches for their solution will be presented. Topics  covered include linear differential equation of order n with coefficients continuous on some interval J,  existence-uniqueness theorem for linear equations of order n, determination of a particular solution of non-homogeneous equations by the method of variation of parameters,  Wronskian matrix of n independent solutions of a homogeneous linear equation,  ordinary and singular points for linear equations of the second order,  solution near a singular point, method of Frobenius, singularities at infinity, simple examples of  Boundary value problems for ordinary linear equation of the second order, Green’s functions, eigenvalues, eigenfunctions, Sturm-Liouville systems, properties of the gamma and beta functions, definition of the gamma function for negative values of the argument; Legendre, Bessel, Chebyshev, Hypergeometic functions and  orthogonality properties.