Introduction & Motivation to Differential Equations, First Order ODE y'=f(x,y), Geometrical interpretation of solution, Equations reducible to separable form, Exact Equations, Integrating factor, Linear Equations, Orthogonal trajectories, Picard's Theorem for IVP (without proof) & Picard's iteration method, Euler's Method, Improved Euler's Method, Elementary types of equations F(x,y,y') = 0 not solved for derivative, Second Order Linear differential equations: Fundamental system of solutions and general solution of homogeneous equation, Use of known solution to find another, Existence and uniqueness of solution of IVP, Wronskian and general solution of non-homogeneous equations, Euler-Cauchy Equation, Extensions of the results to higher order linear equations, Power Series Method - Applications to Legendre Eqn., Legendre Polynomials, Frobenious Method, Bessel equation, Properties of Bessel functions, Sturm-Liouville BVPs, Orthogonal functions, Sturm comparison Theorem.

Introduction to PDE, basic concepts, Linear and quasi-linear first order PDE, second order PDE and classification of second order semi-linear PDE (Canonical form), D'Alemberts formula and Duhamel's principle for one dimensional wave equation, Laplace's and Poisson's equations, Maximum principle with application, Fourier Method for IBV problem for wave and heat equation, rectangular region, Fourier method for Laplace's equation in three dimensions, Numerical Methods for Laplace's and Poisson's equations.