Differential Equations I

1. Ordinary differential Equations of first order: Introduction, initial value problem. The concept of General Solution, Separation of variables, homogeneous equations, linear equations of first order. (Bernoulli and Ricatti equations). Exact equations and integrating factors. Simple applications. (2 weeks)
2. Second order differential equations: linear equations with constant coefficients. Non-homogeneous equations. Variation of parameters. Euler equations. (2 weeks)
3. Newton's equations: Applications in basic problems of Mechanics. Motion under different laws of friction in a homogeneous gravitational field. Harmonic Oscillation with and without friction. Forced oscillations. Motion in a one-dimensional force field. Electrical analogues of mechanics problems. (2 weeks)
4. General study of linear differential equations: The principle of linear combination.The wronskian and its applications. Abel's formula. Calculating the second solution when one is already known. Reduction of order. Complete solution of a non-homogeneous ordinary differential equation when the solutions of the homogeneous one are known. Laplace transformation method for solving linear differential equations with constant coefficients. (3 weeks)
5. Systems of linear differential equations with constant coefficients: method of Elimination and Exponential Substitution method. Applications to problems of coupled oscillators and electrical circuits. Solution methods using matrices. (Evolution operator method.) (2 weeks)
6. Linear differential equations with variable coefficients: Series solutions. Taylor and Frobenius series. Ordinary and Singular points, convergence of Series solutions. Fuchs' theorem. Applications to Bessel, Legendre and Hermite equations. (2 weeks)