Classification of Differential Equations: solutions of ODE, supplementary conditions, direction fields, existence and uniqueness theorems; First Order Differential Equations: linear equations, exact equations, integrating factors, substitutions, and transformations; applications of first-order differential equations, orthogonal and oblique trajectories; Higher Order Linear differential equations: linear differential operators, the basic theory of linear differential equations, homogeneous systems; Methods to Solve Second Order ODEs: reduction of order, nonhomogeneous equation, method of undetermined coefficients, variation of parameters, Euler-Cauchy differential equations, Green’s function; First Order PDEs: complete integral, general solution, Cauchy problems, method of characteristics for linear and quasi-linear equations, Charpit’s method for finding complete integrals, methods for finding general solutions; Second Order PDEs: classifications, reduction to canonical forms, characteristic curves; Boundary Value Problems Related to Linear Equations: applications of Fourier methods, mixed boundary conditions; Problems Involving Cylindrical and Spherical Symmetry: boundary value problems involving special functions, Transform Methods for Boundary Value Problems: Fourier transforms, nonhomogeneous equations, Green’s function.