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Notes on Diffy Qs: Differential Equations for Engineers

Jiří Lebl
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

Notes on Diffy Qs is an introductory textbook on differential equations for engineering students. In approach it is close to (and internally cross-referenced with) books by Edwards and Penney (Differential Equations and Boundary Value Problems) and Boyce and DiPrima (Elementary Differential Equations and Boundary Value Problems). Lebl’s intent is to provide a comparable book in hardcopy at a low cost (or freely available here for download). The hardcopy is entirely in black-and-white whereas the downloadable version has several color illustrations.

The text is intended to support a one-semester or two-quarter course, and could be stretched to two semesters with some supplementary material. The contents are quite standard for a course that aims to introduce both ordinary and partial differential equations. The major topics are first and higher order ordinary differential equations (ODEs), systems of ODEs, then Fourier series and partial differential equations (PDEs), eigenvalue problems (including Sturm-Liouville problems), the Laplace transform and power series methods. Linear algebra is introduced as needed, but is kept to a minimum.

If one considers what knowledge and skills an engineering student should take from an introductory course on differential equations, one might include:

  • Recognizing common differential equations (e.g., variations of harmonic oscillator equation, heat equation, wave equation);
  • Understanding basic techniques for solving ODEs and PDEs;
  • Creating awareness of issues involving existence and uniqueness;
  • Learning rudiments of qualitative behavior of solutions.
  • Getting some experience with numerical solutions.

The current text does reasonably well by these criteria. One might wish for some more depth on qualitative methods, and a clearer message that most differential equations do not have closed form solutions.

The writing is plain but clear throughout and the style is relaxed and conversational. Exercises are plentiful but not particularly distinctive. There are plenty of examples, all carefully worked out. Where examples involve applications, these are all (unsurprisingly) in physics or engineering. Yet it might not be a bad idea for engineers to see at least a few applications outside their field.

Overall, this is a very competent — but not especially inspired – introduction to differential equations, one which is sensitive to the needs of engineering students in future coursework.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

0.1 Notes about these notes
0.2 Introduction to differential equations

1 First order ODEs
1.1 Integrals as solutions
1.2 Slope fields
1.3 Separable equations
1.4 Linear equations and the integrating factor
1.5 Substitution
1.6 Autonomous equations
1.7 Numerical methods: Euler’s method

2 Higher order linear ODEs
2.1 Second order linear ODEs
2.2 Constant coefficient second order linear ODEs
2.3 Higher order linear ODEs
2.4 Mechanical vibrations
2.5 Nonhomogeneous equations
2.6 Forced oscillations and resonance

3 Systems of ODEs
3.1 Introduction to systems of ODEs
3.2 Matrices and linear systems
3.3 Linear systems of ODEs
3.4 Eigenvalue method
3.5 Two dimensional systems and their vector fields
3.6 Second order systems and applications
3.7 Multiple eigenvalues
3.8 Matrix exponentials
3.9 Nonhomogeneous systems

4 Fourier series and PDEs
4.1 Boundary value problems
4.2 The trigonometric series
4.3 More on the Fourier series
4.4 Sine and cosine series
4.5 Applications of Fourier series
4.6 PDEs, separation of variables, and the heat equation
4.7 One dimensional wave equation
4.8 D’Alembert solution of the wave equation
4.9 Steady state temperature and the Laplacian
4.10 Dirichlet problem in the circle and the Poisson kernel

5 Eigenvalue problems
5.1 Sturm-Liouville problems
5.2 Application of eigenfunction series
5.3 Steady periodic solutions

6 The Laplace transform
6.1 The Laplace transform
6.2 Transforms of derivatives and ODEs
6.3 Convolution
6.4 Dirac delta and impulse response

7 Power series methods
7.1 Power series
7.2 Series solutions of linear second order ODEs
7.3 Singular points and the method of Frobenius

Further Reading

Solutions to Selected Exercises