Course information

Introductory notes. Definitions , the solution concept and geometric characteristics. Initial- boundary value problems. Well posed problems. Differential equations of separate variables. Linear, homogeneous, exact differential equations. First order Dif. Eq. : The cases of Riccati,Lagrange and Clairaut dif. equations. Qualitative theory: Existence and uniqueness of solution. Picard and Peano theorems. Linear differential equations: General theory, linear independence of solutions, Wronski determinant. Homogeneous equations with constant coefficients. The nonhomogeneous case: the method of undetermined coefficients and Lagrange method. Power series solution of equations with variable coefficients. Frobenious method. The cases of a regular expansion point and of a regular singular point. The Legendre and Bessel differential equations. Differential systems: The method of eigenvalues and eigenvectors. Laplace transformation: Application to the solution of initial value problems involving ordinary dif. Equations. The Heaviside and Dirac function. Elements of Complex analysis. The concepts of continuity and differentiability. Holomorphic and analytic complex functions. Cauchy - Riemann conditions. Residue theorem. Determination of real integrals via complex analysis. Fourier integrals and applications.