Course description
This course covers eigenvalues and eigenvectors, first order ordinary differential equations,
ordinary linear differential equations of the second order, nonhomogeneous equations with constant coefficients, Cauchy equation, power series solution of differential equations, Laplace transforms, line integral, complex analytic functions, series solution and systems of first order linear differential equations, basic concepts of Partial Differential Equations (PDE), techniques of solutions of first order PDE, Fourier series, second order PDE, vector and tensor analysis.
Course outcomes
Upon the completion of this course, students will be able to:
• Find eigenvalues and eigenvectors of a given square matrix
• Distinguish various classes of differential equations
• State the underlying theory of linear ODEs
• Solve ODEs using various techniques
• Apply the theory of power series to solve certain classes of differential equations
• Apply Laplace transform to solve certain classes of ODE
• Solve system of differential equations
• Express physical problems in terms of differential equations
• Evaluate certain real integrals that are not accessible by real integral calculus
• State the fundamental theorem of line integrals
• Map regions conformally onto another
• Represent periodic functions by using Fourier series
• Solve PDE by using Fourier series
• Find covariant and contravariant components of a vector
Course contents
Click the down arrow icon [ 🔽 ] to expand and collapse the course topics.
🔽 2 h 08 min  The Eigenvalue Problem
 Eigenvalues
 Eigenvectors
 Applications of eigenvalue problems
 Symmetric, skewsymmetric, and orthogonal matrices
 Diagonalization
🔽 1 h 15 min  Ordinary Differential Equations of First Order
 Basic concepts
 Geometric meaning of y’ = f(x, y)
 Direction fields
 Separable ODEs
 Exact ODEs
 Integrating factors
 Linear ODEs
 Bernoulli equation
🔽 1 h 02 min  Second Order Linear Ordinary Differential Equations
 Homogeneous linear ODEs with constant coefficients
 EulerCauchy equations
 Existence and uniqueness of solutions
 Wronskian
 Nonhomogeneous ODEs
🔽 1 h 40 min  Power Series Solution of Differential Equations
 Power series method
 Theory of the power series method
 Legendre’s equation
 Legendre polynomials
 Bessel’s equation
 Bessel functions
🔽 2 h 18 min  Laplace Transforms
 Laplace transform
 Inverse transform
 Linearity, sShifting
 Transforms of derivatives and integrals
 Unit step function, tShifting
 Convolution
 Integral equations
 Differentiation and integration of transforms
 Systems of ODEs
🔽 1 h 15 min  System of Ordinary Differential Equations
 Systems of ODEs
 Basic theory of systems of ODEs
 Constantcoefficient systems
 Phase plane method
 Criteria for critical points
 Stability
 Nonhomogeneous linear systems of ODEs
🔽 1 h 48 min  Complex Integration
 Line Integral in the complex plane
 Cauchy’s integral theorem
 Cauchy’s integral formula
🔽 1 h 04 min  Analytic Functions
 Derivatives of analytic functions
 Evaluation of line integrals
🔽 0 h 58 min  Conformal Mapping
 Conformal mapping
 Linear fractional transformations
 Special linear fractional transformations
🔽 0 h 48 min  Fourier Series
 Fourier series
 Functions of any period p = 2L
 Even and odd functions
 Halfrange expansions
🔽 0 h 54 min  Vector and Tensor Analysis

 Introduction to tensor
 Symmetric and skew symmetric tensors
 Change of basis
 Reciprocal basis
 Covariant and contravariant components of a vector
🔽 1 h 10 min  Partial Differential Equations
 Basic concepts
 Modeling: vibrating string
 Wave equation
 Solution by separating variables
 Use of Fourier series
 D’Alembert’s solution of the wave equation and characteristics
 Heat equation
This course includes:
14 h 30 min recorded video
Downloadable resources (books and articles)
One year access
Access on mobile and TV
Advanced Level
Certificate of completion