Syllabus

Introduction & motivation to differential equations, First Order ODE y'=f(x,y), Geometrical interpretation of simultaneous equations, Equations reducible to separable form, Exact equations, Integrating factor, Linear equations, Orthogonal trajectories, Picard's theorem for IVP & Picard's iteration method, Qualitative properties and theoretical aspects, Euler's method, Elementary types of equations, Second order linear differential equations: fundamental system of solutions and general solution of homogeneous equation, Existence and uniqueness of solution for IVP, Wronskian and general solution of non-homogeneous equations.
Solution of simulation ordinary differential equations using Matrix method (Eigen values and Eigen vectors, Inverse of a matrix). Non-linear first order simultaneous differential equations, Higher order differential equations, Power series method - application to Legendre equation, Legendre polynomials, Frobenius method, Bessel equation. Properties of Bessel functions, Sturm-Liouville BVP, Orthogonal functions, Sturm comparison theorem, Laplace transform, Fourier series and Integrals.

Introduction to PDE, basic concepts, Linear and quasi-Linear first order PDE, 2nd Order semi-linear PDE(Canonical form), D'Alemberts formula and Duhamel's principle for one dimensional wave equation., Laplace and Poisson's equation, Maximum principle with application, Fourier Method for IBV problem for wave and heat equation rectangular region, Fourier Series method for Laplace equation in 3 dimensions.