Advanced Differential Equations

"SYLLABUS- ADVANCED DIFFERENTIAL EQUATIONS, Unit-I Ordinary Differential Equations: Qualitative properties of solution: Oscillation, Wronskian, Sturm separation and comparison theorems. Unit-II Ordinary points, Regular singular points, Frobenius series solution. Unit-III Gauss hyper geometric equation, the point at infinity, Gamma functions, Hermite polynomials. Unit-IV Partial Differential Equations: Origin of first order partial differential equations, Linear equations of the first order, Integral surfaces passing through a given curve, Surface orthogonal to a given system of surface, Non-linear partial differential equations of the first order, Charpits method, Special type of first order equations, Jacobi method. Origin of second order partial differential equation, Linear partial differential equations with constant coefficients, Equations with variable coefficients. Unit-V Problems of Laplace, wave and diffusion equations by the method of separation of variables, Reduction of second order partial differential equation into its canonical form. Non-linear equation of second order. "