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135 pages

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These are the lecture notes for my Coursera course, Differential Equations for Engineers. This course is all about differential equations, and covers material that all engineers should know.

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Description

Content

- Preface

- Introduction to differential equations
- Week I First-order differential equations
- Euler method
- Separable ﬁrst-order equations
- Separable ﬁrst-order equation: example
- Linear ﬁrst-order equations
- Linear ﬁrst-order equation: example
- Application: compound interest
- Application: terminal velocity
- Application: RC circuit
- Week II Second-order differential equations
- Euler method for higher-order odes
- The principle of superposition
- The Wronskian
- Homogeneous second-order ode with constant coefﬁcients
- Case 1: distinct real roots
- Case 2: complex-conjugate roots (Part A)
- Case 2: complex-conjugate roots (Part B)
- Case 3: Repeated roots (Part A)
- Case 3: Repeated roots (Part B)
- Inhomogeneous second-order ode
- Inhomogeneous term: exponential function
- Inhomogeneous term: sine or cosine (Part A)
- Inhomogeneous term: sine or cosine (Part B)
- Inhomogeneous term: polynomials
- Resonance
- Application: RLC circuit
- Application: mass on a spring
- Application: pendulum
- Damped resonance
- Week III The Laplace Transform and Series Solution Methods
- Deﬁnition of the Laplace transform
- Laplace transform of a constant-coefﬁcient ode
- Solution of an initial value problem
- The Heaviside step function
- The Dirac delta function
- Solution of a discontinuous inhomogeneous term
- Solution of an impulsive inhomogeneous term
- The series solution method
- Series solution of the Airy’s equation (Part A)
- Series solution of the Airy’s equation (Part B)
- Week IV Systems of Differential Equations and Partial Differential Equations
- Systems of homogeneous linear ﬁrst-order odes
- Distinct real eigenvalues
- Complex-conjugate eigenvalues
- Coupled oscillators
- Normal modes (eigenvalues)
- Normal modes (eigenvectors)
- Fourier series
- Fourier sine and cosine series
- Fourier series: example
- The diffusion equation
- Solution of the diffusion equation (separation of variables)
- Solution of the diffusion equation (eigenvalues)
- Solution of the diffusion equation (Fourier series)
- Diffusion equation: example

- Appendix
- Appendix A Complex numbers
- Appendix B Nondimensionalization
- Appendix C Matrices and determinants
- Appendix D Eigenvalues and eigenvectors
- Appendix E Partial derivatives
- Appendix F Table of Laplace transforms
- Appendix G Problem solutions

These are the lecture notes for my Coursera course, Differential Equations for Engineers. I cover solution methods for first-order differential equations, second-order differential equations with constant coefficients, and discuss some fundamental applications. I also cover the Laplace transform and series solution methods, systems of linear differential equations, including the very important normal modes problem, and partial differential equations and the method of separation of variables, including the use of Fourier series.

Dr. Jeffrey Chasnov received his BA from UC Berkeley in 1983, and his PhD from Columbia University in 1990. He had postdoctoral appointments at NASA, Stanford University, and the Université Joseph Fourier before expatriating to Hong Kong in 1993, where he is currently a Professor of Mathematics at the Hong Kong University of Science and Technology. Jeff has published over 30 research articles on fluid turbulence, population genetics, and C. elegans biology, and is an experienced lecturer in applied mathematics. One of his favorite courses is Introduction to Differential Equations.