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### Ordinary and Partial Differential Equations and Applications. Instructors: Dr. P. N. Agrawal and Dr. D. N. Pandey, Department of Mathematics, IIT Roorkee. Differential equation are used to express many general laws of nature and have many applications in physical, biological, social, economic, and other dynamical systems. This course contains the existence and uniqueness of solutions of an ODE, homogeneous and non-homogeneous linear systems of differential equations, and power series solutions of second-order homogeneous differential equations.

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### Description

Ordinary and Partial Differential Equations and Applications. Instructors: Dr. P. N. Agrawal and Dr. D. N. Pandey, Department of Mathematics, IIT Roorkee. Differential equation are used to express many general laws of nature and have many applications in physical, biological, social, economic, and other dynamical systems. This course contains the existence and uniqueness of solutions of an ODE, homogeneous and non-homogeneous linear systems of differential equations, and power series solutions of second-order homogeneous differential equations.

Frobenius method, boundary value problems for second order ODE, Green's function, autonomous systems, phase plane, critical points and stability for linear and non-linear systems, eigenvalue problems, Sturm-Liouville problem. Classification of first order PDE, existence and uniqueness of solutions, Nonlinear PDE of first order, Cauchy method of characteristics, Charpit's method, PDE with variable coefficients, canonical forms, characteristic curves, Laplace equation, Poisson equation, wave equation, homogeneous and nonhomogeneous diffusion equation, Duhamel's principle. This course has tremendous applications in diverse fields of Engineering and Sciences such as control theory, numerical analysis and dynamical systems etc. (from **nptel.ac.in**)

### Course Curriculum

- Lecture 01 – Introduction to Differential Equations Unlimited
- Lecture 02 – Initial and Boundary Value Problems with Some Examples Unlimited
- Lecture 03 – Existence and Uniqueness of Solutions of Differential Equations Unlimited
- Lecture 04 – Existence and Uniqueness of Solutions of Differential Equations (cont.) Unlimited
- Lecture 05 – Existence and Uniqueness of Solutions of Differential Equations (cont.) Unlimited
- Lecture 06 – Existence and Uniqueness of Solutions of a System of Differential Equations Unlimited
- Lecture 07 – Linear System Unlimited
- Lecture 08 – Properties of Homogeneous Systems Unlimited
- Lecture 09 – Solution of Homogeneous Linear System with Constant Coefficients Unlimited
- Lecture 10 – Solution of Homogeneous Linear System with Constant Coefficients (cont.) Unlimited
- Lecture 11 – Solution of Homogeneous Linear System with Constant Coefficients (cont.) Unlimited
- Lecture 12 – Solution of Nonhomogeneous Linear System with Constant Coefficients Unlimited
- Lecture 13 – Power Series Unlimited
- Lecture 14 – Uniform Convergence of Power Series Unlimited
- Lecture 15 – Power Series Solution of Second Order Homogeneous Equations Unlimited
- Lecture 16 – Regular Singular Points I Unlimited
- Lecture 17 – Regular Singular Points II Unlimited
- Lecture 18 – Regular Singular Points III Unlimited
- Lecture 19 – Regular Singular Points IV Unlimited
- Lecture 20 – Regular Singular Points V Unlimited
- Lecture 21 – Critical Points Unlimited
- Lecture 22 – Stability of Linear Systems I Unlimited
- Lecture 23 – Stability of Linear Systems II Unlimited
- Lecture 24 – Stability of Linear Systems III Unlimited
- Lecture 25 – Critical Points and Paths of Nonlinear Systems Unlimited
- Lecture 26 – Boundary Value Problems for Second Order Differential Equations Unlimited
- Lecture 27 – Self Adjoint Form Unlimited
- Lecture 28 – Sturm-Liouville Problem and its Properties Unlimited
- Lecture 29 – Sturm-Liouville Problem and its Applications Unlimited
- Lecture 30 – Green’s Function and its Application Unlimited
- Lecture 31 – Green’s Function and its Application (cont.) Unlimited
- Lecture 32 – Origins and Classification of First Order PDEs Unlimited
- Lecture 33 – Initial Value Problem for Quasi-Linear First Order Equations Unlimited
- Lecture 34 – Existence and Uniqueness of Solutions Unlimited
- Lecture 35 – Surfaces Orthogonal to a Given System of Surfaces Unlimited
- Lecture 36 – Nonlinear PDE of First Order Unlimited
- Lecture 37 – Cauchy Method of Characteristics Unlimited
- Lecture 38 – Cauchy Method of Characteristics: Examples Unlimited
- Lecture 39 – Compatible System of First Order Equations Unlimited
- Lecture 40 – Charpit’s Method Unlimited
- Lecture 41 – Charpit’s Method (cont.) Unlimited
- Lecture 42 – Second Order PDE with Variable Coefficients Unlimited
- Lecture 43 – Classification and Canonical Form of Second Order PDE Unlimited
- Lecture 44 – Classification and Canonical Form of Second Order PDE (cont.) Unlimited
- Lecture 45 – Classification and Characteristic Curves of Second Order PDEs Unlimited
- Lecture 46 – Review of Integral Transform: Laplace Transform and its Properties Unlimited
- Lecture 47 – Review of Integral Transform: the Application of Laplace Transform Unlimited
- Lecture 48 – Review of Integral Transform: Fourier Integral Unlimited
- Lecture 49 – Laplace Equation Unlimited
- Lecture 50 – Laplace Equation (cont.) Unlimited
- Lecture 51 – Laplace Equation (cont.) Unlimited
- Lecture 52 – Laplace and Poisson Equations Unlimited
- Lecture 53 – One Dimensional Wave Equations and its Solutions I Unlimited
- Lecture 54 – One Dimensional Wave Equations and its Solutions II Unlimited
- Lecture 55 – One Dimensional Wave Equations and its Solutions III Unlimited
- Lecture 56 – Two Dimensional Wave Equation and its Solution Unlimited
- Lecture 57 – Solution of Nonhomogeneous Wave Equation: Riemann Method Unlimited
- Lecture 58 – Solution of Homogeneous Diffusion Equation Unlimited
- Lecture 59 – Solution of Homogeneous Diffusion Equation (cont.) Unlimited
- Lecture 60 – Duhamel’s Principle Unlimited

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