This course consists of video lectures of full syllabus, exercise solutions and previous year question discussions. Handwritten notes will be provided. Fees is non-refundable. Contact for any queries:- 8882601617 Syllabus:- Unit I: Well posed problems, Existence, uniqueness and continuity of solution of ODEs of first order, Picard’s method, Existence and uniqueness of solution of simultaneous differential equations of first order and ODEs of higher order, Sturm separation and comparison theorems, Homogeneous linear systems, Non-homogeneous linear systems, Linear systems with constant coefficients. Unit II: Two point boundary value problems, Green’s function, Construction of Green’s function, SturmLioville systems, Eigen values and eigen functions, Stability of autonomous system of differential equations, Critical point of an autonomous system and their classification as stable, Asymptotically stable, Strictly stable and unstable, Stability of linear systems with constant coefficients, Linear plane autonomous systems, Perturbed systems, Method of Lyapunov for nonlinear systems. Unit III: Fourier transform and its application to solution of PDEs, Boundary value problems, Maximum and minimum principles, Uniqueness and continuous dependence on boundary data, Solution of the Dirichlet and Neumann problem for a half plane by Fourier transform method, Solution of Dirichlet problem for a circle in form of Poisson integral formula, Theory of Green’s function for Laplace equation in two dimension and application in solution of Dirichlet and Neumann problem for half plane and circle, Theory of Green’s function for Laplace equation in three dimension and application in solution of Dirichlet and Neumann problem for semi-infinite spaces and spheres. Unit IV: Wave equation, Helmholtz’s first and second theorems, Green’s function for wave equation, Duhamel’s principles for wave equation, Diffusion equation, Solution of initial boundary value problems for diffusion equation, Green’s function for diffusion equation, Duhamel’s principles for heat equation.