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Meaning of differential equations, order, degree, solutions, etc., of differential equations.

Analytical solutions of first-order ordinary differential equations, such as variable-separable equations, homogeneous equations, non-homogeneous equations convertible to homogeneous forms, exact differential equations, inexact differential equations, linear differential equations, Bernoulli equation, and Riccati equation.

Applications of first-order ordinary differential equations.

Analytical solutions of homogeneous linear second-order ordinary differential equations with constant coefficients.

Analytical solutions of non-homogeneous linear second-order ordinary differential equations by the methods of undetermined coefficients and variation of parameters.

Applications of second-order ordinary differential equations to mechanical vibrations and electric circuits.

Analytical solutions of some higher-order ordinary differential equations by reduction to first-order ordinary differential equations.

Series solutions of second-order linear differential equations with variable coefficients - Frobenius method, gamma functions and Bessel's equations, Legendre's equations.

Solutions of second-order boundary-value problems - eigenvalue problems and Sturm-Liouville problems.