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Differential Equations and Applications · LearnSpace
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Differential Equations and Applications

Курс от Birla Institute of Technology & Science, Pilani
Средний≈ 27.8 чАнглийский
О курсеНавыкиПрограммаПреподаватели

О курсе

This course provides a comprehensive study of ordinary differential equations (ODEs), focusing on both theoretical concepts and solution techniques. It begins with the classification and solution of linear and non-linear first-order ODEs, followed by an in-depth exploration of second-order ODEs, including the concepts of linear dependence and independence of solutions. Various methods for solving second-order equations are introduced, such as the method of undetermined coefficients, variation of parameters, and reduction of order. The course also covers advanced techniques like the Laplace Transform for solving initial value problems and the application of Fourier series for representing periodic solutions. Sturm-Liouville problems are studied as a framework for solving boundary value problems and understanding eigenfunction expansions. Additionally, the course addresses the solution of systems of linear ODEs using matrix methods and eigenvalue analysis. This course equips students with the mathematical tools necessary to model and analyze real-world dynamic systems in science and engineering.

Навыки, которые вы освоите

Differential EquationsApplied MathematicsAdvanced MathematicsLinear AlgebraMathematical ModelingDependency AnalysisMathematical Theory & AnalysisDerivativesEngineering AnalysisCalculusEngineering CalculationsIntegral Calculus

Программа курса

10 модулей · 258 учебных материалов

01Introduction to Differential Equations24 материалов

Course Introduction

Course OverviewЧтениеCourse & Instructor - Introductory Video Видео

Introduction to Differential Equations

IntroductionВидеоQuiz: Introduction

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BITS Pilani Instructors Group

Преподаватель курса

Differential Equations and Applications
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Обучение на Coursera

≈ 27.8 ч

10 модулей

Язык: Английский

Часть программы вашего университета
Задание
What is a Differential Equation?Видео
Quiz: What is a Differential Equation?Задание
Order of a Differential EquationВидео
Quiz: Order of a Differential EquationЗадание
Degree of a Differential EquationВидео
Quiz: Degree of a Differential EquationЗадание
Classification of Differential EquationsВидео
Quiz: Classification of Differential EquationsЗадание
Introduction to Differential EquationsЧтение

Solution of a Differential Equation

Applications of Differential EquationsВидеоQuiz: Applications of Differential EquationsЗаданиеIntroductionВидеоQuiz: IntroductionЗаданиеSolution of a Differential EquationВидеоQuiz: Solution of a Differential EquationЗаданиеFormation of Differential EquationВидеоQuiz: Formation of Differential EquationЗаданиеExampleВидеоQuiz: ExampleЗаданиеSolution of a Differential EquationЧтение
02First order Differential Equations20 материалов

Linear differential equations constant coefficients

IntroductionВидеоQuiz: IntroductionЗаданиеVariables SeparableВидеоQuiz: Variables SeparableЗаданиеHomogeneous EquationsВидеоQuiz: Homogeneous EquationsЗаданиеDifferential Equations Reducible to HomogeneousВидеоQuiz: Differential Equations Reducible to HomogeneousЗаданиеLinear Differential equationsВидеоQuiz: Linear Differential equationsЗаданиеBernoulli’s equationВидеоQuiz: Bernoulli’s equationЗаданиеExact Differential EquationВидеоQuiz: Exact Differential EquationЗаданиеNon -Exact Differential EquationВидеоQuiz: Non -Exact Differential EquationЗаданиеNon -Exact Differential Equation-solutionВидеоQuiz: Non -Exact Differential Equation-solutionЗаданиеLinear differential equations constant coefficientsЧтение

Summary and Assessment

Graded Quiz for Week 1 and 2Задание
03Applications of first order Differential equations17 материалов

Applications of first order differential equations

IntroductionВидеоQuiz: IntroductionЗаданиеGrowth and DecayВидеоQuiz: Growth and DecayЗаданиеNewton’s law of coolingВидеоQuiz: Newton’s law of coolingЗаданиеDynamics of tumour growthВидеоQuiz: Dynamics of tumour growthЗаданиеElectric circuitsВидеоQuiz: Electric circuitsЗаданиеOrthogonal TrajectoriesВидеоQuiz: Orthogonal TrajectoriesЗаданиеOrthogonal TrajectoriesВидеоQuiz: Orthogonal TrajectoriesЗаданиеEconomicsВидеоQuiz: EconomicsЗаданиеApplications of first order differential equationsЧтение
04Higher order Linear Differential equations with constant coefficients23 материалов

Homogeneous Equations

IntroductionВидеоQuiz: IntroductionЗаданиеSolutionВидеоQuiz: SolutionЗаданиеExample -1ВидеоQuiz: Example -1ЗаданиеExample -2ВидеоQuiz: Example -2ЗаданиеHomogeneous EquationsЧтение

Non – Homogeneous Equations

General and Particular SolutionВидеоQuiz: General and Particular SolutionЗаданиеParticular solution with eaxВидеоQuiz: Particular solution with eaxЗаданиеParticular solution with sin(bx) or Cos(bx)ВидеоQuiz: Particular solution with sin(bx) or Cos(bx)Задание

Summary and Assessment

Graded Quiz for Week 3 and 4Задание
05Laplace Transform Methods27 материалов

Laplace Transformation

IntroductionВидеоQuiz: IntroductionЗаданиеDefinitionВидеоQuiz: DefinitionЗаданиеProperties of Laplace TransformВидеоQuiz: Properties of Laplace TransformЗаданиеExamplesВидеоQuiz: ExamplesЗаданиеExamplesВидеоQuiz: ExamplesЗаданиеSpecial FunctionsВидеоQuiz: Special FunctionsЗаданиеLaplace TransformationЧтение

Inverse Laplace Transforms

DefinitionВидеоQuiz: DefinitionЗаданиеExamplesВидеоQuiz: ExamplesЗаданиеPropertiesВидеоQuiz: PropertiesЗаданиеExamplesВидео

Applications of Laplace Transforms

ExamplesВидеоQuiz: ExamplesЗаданиеConvolution TheoremsВидеоQuiz: Convolution TheoremЗаданиеApplications of Laplace TransformsЧтение
06Solution of ODE using Laplace transforms33 материалов

Solution of Homogeneous ODE using Laplace Transform

IntroductionВидеоQuiz: IntroductionЗаданиеSolving First-Order Homogeneous ODEs Using Laplace TransformВидеоQuiz: Solving First-Order Homogeneous ODEs Using Laplace TransformЗаданиеSolving Second-Order Homogeneous ODEs Using Laplace TransformВидеоQuiz: Solving Second-Order Homogeneous ODEs Using Laplace TransformЗаданиеSolving Homogeneous ODEs involving partial fractions with distinct real rootsВидеоQuiz: Solving Homogeneous ODEs involving partial fractions with distinct real rootsЗаданиеSolving Homogeneous ODEs involving partial fractions with repeated rootsВидеоQuiz: Solving Homogeneous ODEs involving partial fractions with repeated rootsЗаданиеSolving Homogeneous ODEs involving irreducible quadratic equationsВидеоQuiz: Solving Homogeneous ODEs involving irreducible quadratic equationsЗаданиеSolution of Homogeneous ODE using Laplace TransformЧтение

Solution of Non-homogeneous ODE using Laplace Transform

Steps to SolveВидеоQuiz: Steps to SolveЗаданиеSolving Non-Homogeneous ODE with f(t) = tneatВидеоQuiz: Solving Non-Homogeneous ODE with f(t) = tneatЗаданиеSolving boundary value problem using Laplace transform methodВидеоQuiz: Solving boundary value problem using Laplace transform methodЗадание

Non-Homogeneous ODE with Heaviside Functions

Non-homogeneous ODE with Heaviside functionsВидеоQuiz: Non-homogeneous ODE with Heaviside functionsЗаданиеNon-Homogeneous ODE with Delta functionsВидеоQuiz: Non-Homogeneous ODE with Delta functionsЗаданиеNon-Homogeneous ODE using convolution theoremВидеоQuiz: Non-Homogeneous ODE using convolution theoremЗадание

Summary and Assessment

Module Wrap Up VideoВидеоGraded Quiz for Week 5 and 6Задание
07Introduction to Fourier Series24 материалов

Periodic functions and series representation

Definition of Periodic FunctionВидеоQuiz: Definition of Periodic FunctionЗаданиеPiecewise and Discontinuous Periodic FunctionsВидеоQuiz: Piecewise and Discontinuous Periodic FunctionsЗаданиеProperties of Periodic FunctionВидеоQuiz: Properties of Periodic FunctionЗаданиеPeriodic functions and series representationЧтение

Euler Formula

Orthogonality of Trigonometric FunctionsВидеоQuiz: Orthogonality of Trigonometric FunctionsЗаданиеDerivation of Euler’s FormulaВидеоQuiz: Derivation of Euler’s FormulaЗаданиеComputation of Fourier Series for an ExampleВидеоQuiz: Computation of Fourier Series for an ExampleЗадание

Convergence of Fourier Series

Sum of a Fourier SeriesВидеоQuiz: Sum of a Fourier SeriesЗаданиеProof of Convergence TheoremВидеоQuiz: Proof of Convergence TheoremЗаданиеExample of Convergent Fourier SeriesВидеоQuiz: Example of Convergent Fourier SeriesЗадание

Summary and Assessment

Module Wrap Up VideoВидео
08Extension and Applications of Fourier Series30 материалов

Fourier Series of functions with arbitrary period, odd and even functions

Extension of Fourier Series to Function with Arbitrary PeriodВидеоQuiz: Extension of Fourier Series to Function with Arbitrary PeriodЗаданиеExample of a Function with Arbitrary PeriodВидеоQuiz: Example of a Function with Arbitrary PeriodЗаданиеEven and Odd FunctionsВидеоQuiz: Even and Odd FunctionsЗаданиеFourier Series of Even and Odd FunctionsВидеоQuiz: Fourier Series of Even and Odd FunctionsЗаданиеExample of Fourier Series of Even FunctionВидеоQuiz: Example of Fourier Series of Even FunctionЗаданиеExample of Fourier Series of Odd FunctionВидеоQuiz: Example of Fourier Series of Odd FunctionЗаданиеFourier Series of functions with arbitrary period, odd and even functionsЧтение

Half Range Expansion and Application

Concept of Half Range ExpansionВидеоQuiz: Concept of Half Range ExpansionЗаданиеExample of Even Periodic ExtensionВидеоQuiz: Example of Even Periodic ExtensionЗаданиеExample of Odd Periodic ExtensionВидеоQuiz: Example of Odd Periodic ExtensionЗадание

Parseval’s Identity

Idea of Parseval’s IdentityВидеоQuiz: Idea of Parseval’s IdentityЗаданиеApplication of Parseval’s IdentityВидеоQuiz: Application of Parseval’s IdentityЗаданиеParseval’s IdentityЧтение

Summary and Assessment

Module Wrap Up VideoВидеоGraded Quiz for Week 7 and 8Задание
09Sturm Liouville Equation and its Applications26 материалов

Sturm Liouville Problem

Definition of Sturm Liouville Equation and ProblemВидеоQuiz: Definition of Sturm Liouville Equation and ProblemЗаданиеBoundary value problemВидеоQuiz: Boundary value problemЗаданиеEigenvalue and EigenfunctionВидеоQuiz: Eigenvalue and EigenfunctionЗаданиеLagrange’s IdentityВидеоQuiz: Lagrange’s IdentityЗаданиеSturm Liouville ProblemЧтение

Orthogonal Functions and Application

Orthogonality of Eigenfunctions with respect to a functionВидеоQuiz: Orthogonality of Eigenfunctions with respect to a functionЗаданиеProof of Orthogonality of Eigen functions of Sturm Liouville ProblemВидеоQuiz: Proof of Orthogonality of Eigen functions of Sturm Liouville ProblemЗаданиеSturm–Liouville Problem as a Self-Adjoint OperatorВидеоQuiz: Sturm–Liouville Problem as a Self-Adjoint OperatorЗадание

Generalized Fourier series, Legendre series, Bessel series

Generalized Fourier seriesВидеоQuiz: Generalized Fourier seriesЗаданиеFourier Legendre SeriesВидеоQuiz: Fourier Legendre SeriesЗаданиеFourier Bessel SeriesВидеоQuiz: Fourier Bessel SeriesЗадание

Summary and Assessment

Module Wrap Up VideoВидео
10Linear System of ODE34 материалов

Linear System of ODE and uncoupled system

Definition of Linear system of ODEВидеоQuiz: Definition of Linear system of ODEЗаданиеUncoupled linear systemВидеоQuiz: Uncoupled linear systemЗаданиеMethod to solve uncoupled linear systemВидеоQuiz: Method to solve uncoupled linear systemЗаданиеExample of an uncoupled linear systemВидеоQuiz: Example of an uncoupled linear systemЗаданиеLinear System of ODE and uncoupled systemЧтение

Diagonalization

Eigenvalues and eigenvectorsВидеоQuiz: Eigenvalues and eigenvectorsЗаданиеDiagonalization of a matrixВидеоQuiz: Diagonalization of a matrixЗаданиеExample of Linear system of ODE with A as a diagonal matrixВидеоQuiz: Example of Linear system of ODE with A as a diagonal matrixЗадание

Solution of system of linear ODE

Definition of eAt and deAtdtВидеоQuiz: Definition of eAt and deAtdtЗаданиеSome results about eAtВидеоQuiz: Some results about eAtЗаданиеFundamental theorem for Linear systemВидеоQuiz: Fundamental theorem for Linear systemЗадание

Summary and Assessment

Module Wrap Up VideoВидеоGraded Quiz for Week 9 and 10Задание

Course Wrap-Up

Course SummaryЧтение
Particular solution with polynomialsВидео
Quiz: Particular solution with polynomialsЗадание
Particular solution with eaxVВидео
Quiz: Particular solution with eaxVЗадание
ExamplesВидео
Quiz: ExamplesЗадание
Non – Homogeneous EquationsЧтение
Quiz: ExamplesЗадание
Inverse Laplace TransformsЧтение
Solving an ODE in LC-circuit using Laplace transform methodВидео
Quiz: Solving an ODE in LC-circuit using Laplace transform methodЗадание
Solution of Non-homogeneous ODE using Laplace TransformЧтение
Non-homogeneous ODE with Integral forcing functionВидео
Quiz: Non-homogeneous ODE with Integral forcing functionЗадание
Non-Homogeneous ODE with Heaviside FunctionsЧтение
Euler FormulaЧтение
Gibbs PhenomenonВидео
Quiz: Gibbs PhenomenonЗадание
Convergence of Fourier SeriesЧтение
Application of Fourier Series to Solve ODEВидео
Quiz: Application of Fourier Series to Solve ODEЗадание
Application of Fourier Series to solve an ODE ExampleВидео
Half Range Expansion and ApplicationЧтение
Application of Orthogonal EigenfunctionsВидео
Quiz: Application of Orthogonal EigenfunctionsЗадание
Orthogonal Functions and ApplicationЧтение
Generalized Fourier series, Legendre series, Bessel seriesЧтение
General solution of x=Ax where A is a diagonal matrixВидео
Quiz: General solution of x=Ax where A is a diagonal matrixЗадание
General solution of x=Ax where A is a diagonal matrixВидео
Quiz: General solution of x=Ax where A is a diagonal matrixЗадание
DiagonalizationЧтение
Solving Initial Value ProblemВидео
Quiz: Solving Initial Value ProblemЗадание
Solving second order differential equation as a linear systemВидео
Quiz: Solving second order differential equation as a linear systemЗадание
Solution of system of linear ODEЧтение