Course information

Preliminaries. Origin and usefulness. Mathematical models. The notion and classification of differential equations. The notion of solution, initial-boundary value problems. Well posed problems. First order differential equations. Linear, separation of variables, exact, integrating factors, homogeneous, Bernoulli equation, Riccati equation, existence and uniqueness solution theorem, modelling of physical problems. Linear differential equations of second and higher order.General theory of homogenous equations, linear independent solutions and Wronskian determinant, Abel’s theorem, reduction of order-D’Alembert’s method, non-homogeneous equations, method of variations of parameters, equations with constant coefficients, characteristic polynomial, simple, multiple and complex roots, method of undetermined coefficients. Laplace transform. Definition, solution to initial value problems, Heavyside and Dirac functions, equations with non-continuous non-homogeneous term, convolution theorem, Volterra type equations. Differential equation systems of first order. Homogeneous linear systems with constant coefficients, complex, multiple eigenvalues, the phase plane, autonomous systems and stability, non-homogeneous linear systems. Solving linear second order equations with power series method. Power series solution about a regular point, Legendre equation, Legendre polynomial, Euler equation, solutions in the neighborhood of regular singular points, Bessel equation. Trigonometric Fourier series. Fourier coefficients, convergence theorem, even and odd functions, sine and cosine expansions, complex form of Fourier series. Boundary value problems. Homogeneous Sturm-Liouville problems, eigenvalues and eigensolutions. Separation of variables. The wave equation, vibration of an elastic string, the D’ Alembert’s solution, the method of separation of variables in two and three dimensions, modelling of physical problems.