Differential Equations for Engineers

Week 1: Introduction to Differential Equations, Solutions of first order ODEs, Homogeneous Equations, Exact Equations. Week 2: Solution methods for first order ODEs, Reducible to Exact Equations, Integrating factors, Linear first order ODE, Reducible to linear equations, Bernoulli’s Equation. Week 3: Introduction to Second order ODEs' , Properties of solutions of second order homogeneous ODEs', Abel's formula to find the other linear independent solution, Abel's formula-Demonstration with examples. Week 4: Second order ODE's with constant coefficients, Euler-Cauchy equation, Non-homogeneous ODEs-Variation of parameters, Method of undetermined coefficients,Demonstration of Method of undetermined coefficients Week 5: Power series and its properties, Power series method to solve second order linear homogeneous ODEs, Legendre Equation, Properties of Legendre polynomials. Week 6: Classification of Singular points, Solution around a regular singular point- Frebeneous Method, Bessel Equation and properties of Bessel functions. Week 7: Sturm-Liouville theory, Finding Eigenvalues and Eigenfunctions, Generalised Fourier series. Week 8: Introduction to second order Linear Partial Differential Equations (PDEs), Classification of nd order linear PDEs, Solutions by the method of classification. Week 9: One-dimensional Wave Equation, D’Alembert’s solution, Solution of wave equation in semi-infinte domains, Uniqueness by the energy argument, non-homogeneous wave equation and its solution. Week 10: Seperation of variable method for -dim wave equation over a finite domain, Vibration of a finite string, Two-Dimensional Wave equation, Vibration of a drum. Week 11: One-dimensional Heat equation, Temperature distributions in an infinite,semi-infinite and finite rods, Uniqueness of solutions, Solution of a Heat equation with external source Week 12: Steady-State Heat Equation, Solutions of the Laplace equation in rectangular domains, Solution of the Laplace equation in circular domains.