Differential Equations for Engineers Course

Lesson 1 • Mathematical Prerequisites Review

Reviews calculus and algebra tools required for solving ODEs. Ensures students can differentiate, integrate, and manipulate expressions fluently.

Lesson 2 • What Are Differential Equations

Introduces the definition of a differential equation and its components. Establishes vocabulary used throughout the course.

Lesson 3 • Initial and Boundary Value Problems

Distinguishes IVPs from BVPs and states existence-uniqueness conditions. Connects problem setup to physical engineering constraints.

Lesson 4 • Classification and Terminology

Categorises ODEs by linearity, homogeneity, and autonomy. Correct classification guides solution strategy selection.

Lesson 1 • Separable Equations

Teaches separation of variables as the most direct first-order technique. Students apply it to exponential growth, decay, and mixing problems.

Lesson 2 • Bernoulli and Riccati Equations

Reduces nonlinear Bernoulli and Riccati equations to linear form via substitution. Broadens the toolkit for nonlinear first-order problems.

Lesson 3 • Linear First-Order Equations

Introduces the integrating factor method for linear ODEs. Connects the technique to RC circuit and fluid-flow engineering models.

Lesson 4 • Exact Equations and Integrating Factors

Identifies exact equations via the exactness condition and constructs potential functions. Extends to non-exact cases using integrating factors.

Lesson 1 • Nonhomogeneous Equations: Undetermined Coefficients

Applies the method of undetermined coefficients to find particular solutions. Covers polynomial, exponential, and sinusoidal forcing functions.

Lesson 2 • Cauchy-Euler Equations

Solves variable-coefficient Cauchy-Euler equations via power substitution. Connects to beam deflection and fluid mechanics applications.

Lesson 3 • Variation of Parameters

Differentiates particular solutions for arbitrary forcing functions using variation of parameters.

Lesson 4 • Mechanical Vibrations Modelling

Translates spring-mass-damper systems into second-order ODEs. Students interpret free, damped, and forced vibration responses physically.

Lesson 1 • Definition and Basic Properties

Defines the Laplace transform via the improper integral and establishes linearity. Students compute transforms of standard engineering functions.

Lesson 2 • Step Functions and Discontinuous Forcing

Models piecewise forcing using the Heaviside step function and the second shifting theorem. Applies to switched electrical and mechanical systems.

Lesson 3 • Inverse Laplace Transform

Recovers time-domain solutions using partial fractions and transform tables. Covers simple poles, repeated poles, and complex poles.

Lesson 4 • Transform of Derivatives and IVPs

Differentiates transform rules for first and higher derivatives.

Lesson 1 • Phase Plane Analysis

Classifies equilibria of two-dimensional linear systems using trace and determinant. Students sketch phase portraits and interpret system behaviour.

Lesson 2 • Converting Higher-Order ODEs to Systems

Rewrites nth-order ODEs as first-order systems via state variables. Establishes the matrix form used throughout the chapter.

Lesson 3 • Nonhomogeneous Systems

Extends variation of parameters to systems with forcing terms. Students solve driven coupled oscillator and circuit problems.

Lesson 4 • Matrix Exponential and Fundamental Matrix

Defines the matrix exponential and fundamental solution matrix. Provides a unified framework for both homogeneous and nonhomogeneous systems.

Lesson 1 • Sturm-Liouville Theory Overview

Frames eigenvalue problems in Sturm-Liouville form and establishes orthogonality of eigenfunctions. Prepares students for Fourier series and PDE expansions.

Lesson 2 • Bessel's Equation and Bessel Functions

Differentiates Bessel functions of the first and second kind from Bessel's equation.

Lesson 3 • Legendre's Equation and Polynomials

Solves Legendre's equation and identifies polynomial solutions. Connects to spherical coordinate problems in electromagnetics and fluid flow.

Lesson 4 • Frobenius Method at Regular Singular Points

Applies the Frobenius method when the origin is a regular singular point. Handles indicial equation roots that differ by an integer.

Lesson 1 • Classification of Second-Order PDEs

Classifies PDEs as elliptic, parabolic, or hyperbolic using the discriminant. Links each class to physical phenomena and appropriate solution methods.

Lesson 2 • Laplace Equation and Steady-State Problems

Solves the two-dimensional Laplace equation on rectangular and circular domains. Models steady-state temperature and electrostatic potential distributions.

Lesson 3 • Fourier Series and Orthogonal Expansions

Represents periodic and piecewise functions as Fourier series. Establishes the tool needed for PDE solution by eigenfunction expansion.

Lesson 4 • Heat Equation: Separation of Variables

Solves the one-dimensional heat equation with homogeneous boundary conditions. Students interpret temperature distribution evolution over time.

Lesson 1 • Numerical Methods for Systems and BVPs

Extends Runge-Kutta to first-order systems and applies shooting and finite difference methods to BVPs. Covers two-point boundary value problems.

Lesson 2 • Finite Difference Methods for PDEs

Discretises the heat and wave equations using finite differences. Students analyse stability via the Courant-Friedrichs-Lewy condition.

Lesson 3 • Euler and Improved Euler Methods

Derives Euler's method from the Taylor expansion and improves it with the predictor-corrector approach. Students quantify local and global truncation error.

Lesson 4 • Multistep Methods

Introduces Adams-Bashforth and Adams-Moulton methods as efficient alternatives to single-step solvers. Analyses consistency, stability, and convergence.