1. Introduction: Order and degree of ordinary differential equations (ODEs); Simple physical examples.
2. Solution of first-order ODEs (general): Separation of variables; Homogeneous equations (HEs); Linear HEs; Reduction to linear HEs; Exact equations; Integration factor. Application: Family of curves orthogonal to a given family.
3. General linear first-order ODEs; Bernoulli and Riccati equations.
Clairaut equation: General and "envelope" solutions.
4. Linear second-order ODEs (LSODEs) with constant coefficients: Homogeneous case; Solution of inhomogeneous case by two-step integration or by other systematic methods and educated guesses. Euler equation.
5. Other second-order ODEs: Equations "without x" or "without y".
6. Theory of general LSODEs: Independent solutions of homogeneous case; Wronskian; Abel's theorem; Method of "variation of parameters".
7. Classic LSODEs of Mathematical Physics: Airy, Bessel, Chebyshev, Hermite, Laguerre, and Legendre equations. Series solution of general LSODEs: Taylor expansion around regular point; Frobenius expansion around regular singular point; Different cases of characteristic indices; Convergence.
8. Introduction to the Calculus of Variations; Isoperimetric problems and their solution by Lagrange-multipliers method.