The meaning and need for numerical methods.

Definition and use

Introduction

Numerical solutions of equations in one variable.

Meaning and examples of equations in one variable, meaning root of an equation and zero of a function.

Worked examples on the bisection method of solution of equations in one variable.

Worked examples I

More worked examples on Newton's method of solution of equations in one variable.

Worked examples III

Worked examples II

Bisection method of solution of equations in one variable.

Bisection method

Newton's method of solution of equations in one variable.

Newton's method

Condition for the existence of a solution within an interval.

Existence of a solution

Advantages and disadvantages of Newton's method solution of equations in one variable.

Overview of Newton's method

Advantages and disadvantages of the bisection method of solution of equations in one variable.

Overview of the bisection method

More worked examples on the bisection method of solution of equations in one variable.

Equations in One Variable

Introduction to numerical methods  for interpolation and polynomial approximation.

Interpolation and Polynomial Approximation

Numerical differentiation using finite differences.

Introduction to numerical methods  for differentiation.

Differentiation

Numerical integration of single-variable functions.

An overview of the theory of numerical integration methods.

More worked examples on numerical integration.

Worked examples IV

Worked examples on integration by the trapezoidal rule.

Simpson's rule

Trapezoidal rule

Worked examples on integration by Simpson's 1/3 rule.

Integration

Numerical solutions of initial-value problems.

Introduction to numerical methods  for ordinary differential equations.

Ordinary Differential Equations

Numerical solution of non-linear equations, integration.

Numerical Methods - Advanced Calculus

Meaning of differential equations, order, degree, solutions, etc., of differential equations.

Meaning of the solution of differential equations.

Solution of differential equations

Meaning of initial-value problems and boundary-value problems.

Initial-value and boundary-value problems

Classification of differential equations into ordinary and partial differential equations.

Ordinary and partial differential equations

Worked examples on boundary-value and initial-value problems.

Worked examples (2)

Meaning of the order of differential equations.

Order of differential equations

Definition

Identifying linear and non-linear differential equations.

Linear and non-linear differential equations

Identifying the degree of differential equations.

Degree of differential equations

Worked examples on identifying differential equations and their solutions.

Worked examples (1)

An overview of common notations for derivatives.

Notations

Applications of first-order ordinary differential equations.

Modelling rate of chemical reactions with first-order ordinary differential equations.

Rate of chemical reactions

Modelling temperature change problems with first-order ordinary differential equations.

Newton's law of cooling

Modelling population growth with first-order ordinary differential equations.

Population growth

Determining the oblique trajectories of a family of curves.

Oblique trajectories

Worked examples on Newton's second law of motion.

Newton's second law of motion

Determining the orthogonal trajectories of a family of curves.

Orthogonal trajectories

Modelling radioactive decay with first-order ordinary differential equations.

Radioactive decay

More examples on applications of first-order ordinary differential equations.

More worked examples (1)

Modelling electric circuit problems with first-order ordinary differential equations.

Electric circuits

Modelling exponential growth and decay with first-order ordinary differential equations.

Exponential growth and decay

Applications of First-Order Ordinary Differential Equations

Analytical solutions of homogeneous linear second-order ordinary differential equations with constant coefficients.

How to obtain the auxiliary equation of homogeneous equations with constant coefficients.

Solving homogeneous equations with constant coefficients (1)

More worked examples on solving homogeneous linear ordinary differential equations with constant coefficients.

Understanding linear dependence of functions and the Wronskian.

Linear dependence

Meaning of homogeneous and non-homogeneous linear differential equations.

Linearly-independent solutions and the general solution of homogeneous linear second-order differential equations.

General solution of homogeneous equations

How to obtain the general solution of homogeneous equations with constant coefficients from the roots of the auxiliary equation.

Solving homogeneous equations with constant coefficients (2)

General solution of non-homogeneous linear second-order differential equations.

General solution of non-homogeneous equations

Solutions of Second-Order Ordinary Differential Equations (1)

Analytical solutions of first-order ordinary differential equations, such as variable-separable equations, homogeneous equations, non-homogeneous equations convertible to homogeneous forms, exact differential equations, inexact differential equations, linear differential equations, Bernoulli equation, and Riccati equation.

Solution of Bernoulli's ordinary differential equation.

Simple non-linear equations (1)

Solution of homogeneous differential equations.

Homogeneous equations

More worked examples on solution of first-order ordinary differential equations.

Solution linear differential equations with integrating factors.

Linear differential equations

An introduction to first-order differential equations and their solutions.

Solution of non-homogeneous differential equations reducible to homogeneous form.

Non-homogeneous equations

Solution of Riccati's ordinary differential equation.

Simple non-linear equations (2)

Solution of variable-separable differential equations.

Variable-separable equations

More worked examples (2)

More worked examples (3)

Inexact differential equations and their solutions with integrating factors.

Inexact differential equations

Solution of exact differential equations.

Exact differential equations

Solutions of First-Order Ordinary Differential Equations

Applications of second-order ordinary differential equations to mechanical vibrations and electric circuits.

Application of second-order ordinary differential equations to problems on vibrating springs.

Vibrating springs

Application of second-order ordinary differential equations to problems on damped vibrations.

Damped vibrations

Application of second-order ordinary differential equations to problems on forced vibrations.

Forced vibrations

Application of second-order ordinary differential equations to problems on electric circuits.

More worked examples on applications of second-order ordinary differential equations.

Applications of Second-Order Ordinary Differential Equations

Analytical solutions of some higher-order ordinary differential equations by reduction to first-order ordinary differential equations.

An introduction to the technique of reducing some higher-order ODEs to first order.

Solutions of Second-Order Ordinary Differential Equations (3)

Series solutions of second-order linear differential equations with variable coefficients - Frobenius method, gamma functions and Bessel's equations, Legendre's equations.

Introduction to the general series solution method for second-order linear differential equations.

Solutions of Second-Order Ordinary Differential Equations (4)

Solutions of second-order boundary-value problems - eigenvalue problems and Sturm-Liouville problems.

An introduction to second-order boundary-value problems and their solution technique.

Second-Order Boundary-Value Problems

Analytical solutions of non-homogeneous linear second-order ordinary differential equations by the methods of undetermined coefficients and variation of parameters.

More worked examples on the method of undetermined coefficients.

Worked examples (3)

An overview of methods of determining the particular integral.

Variation of parameters

Worked examples on the method of variation of parameters.

Worked examples (4)

Worked examples (5)

More worked examples on the method of variation of parameters.

Worked examples (6)

Worked examples (7)

Worked examples on the method of undetermined coefficients.

Solution of non-homogeneous second-order linear ordinary differential equations by the method of undetermined coefficients.

Undetermined coefficients

Solutions of Second-Order Ordinary Differential Equations (2)

Ordinary Differential Equations - Advanced Calculus

Integral transforms (Laplace, Fourier, Mellin, Hankel and other transforms ), their inverses and applications.

Mastering Integral Transforms: Laplace, Fourier and Others

Advanced engineering mathematics covering numerical and analytical methods of engineering analysis.

Curated for third-year students of engineering at Obafemi Awolowo University, Ile-Ife, Nigeria. Students and professionals with similar learning goal will also find this learning track useful.

Tutorials for undergraduates in engineering and physical sciences.

CHE 306: Engineering Analysis II

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