--- title: Laplace Transforms description: The Laplace transform converts differential equations into simpler algebraic equations. Learners will master transform techniques, solve linear ordinary differential equations with constant coefficients, and analyze discontinuous or impulsive forcing functions in systems. category: mathematics subcategory: differential-equations difficulty: beginner, intermediate, advanced url: /subject/laplace-transforms --- # Laplace Transforms The Laplace transform converts differential equations into simpler algebraic equations. Learners will master transform techniques, solve linear ordinary differential equations with constant coefficients, and analyze discontinuous or impulsive forcing functions in systems. ## Available Resources 4 Books • 3 Courses • 2 Websites ## Courses ### 1. Introduction to Differential Equations Learn differential equations: explore methods for solving, modeling, and analyzing them. Taught by the Khan Academy Team. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/differential-equations **Tags:** differential-equations, ordinary-differential-equations, separable-equations, laplace-transform, khan-academy ### 2. MIT 18.03SC Differential Equations, Unit III: Fourier Series and Laplace Transform **Author:** Arthur Mattuck, Haynes Miller, David Jerison, Jennifer French, Jeremy Orloff Self-paced MIT unit with Arthur Mattuck's lecture videos, recitation videos, course notes, quizzes, and problem sets with solutions covering step and delta functions, convolution, the Laplace transform, partial fractions, solving IVPs, transfer functions, and poles. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/pages/unit-iii-fourier-series-and-laplace-transform/ **Tags:** laplace-transform, fourier-series, differential-equations, transfer-functions, convolution ### 3. Differential Equations for Engineers This course is all about differential equations and covers both theory and applications. In the first five weeks, students will learn about ordinary differential equations, while the sixth week is an introduction to partial differential equations. The course includes 56 concise lecture videos, with a few problems to solve after each lecture. After each major topic, there is a short practice quiz. At the end of each week, there is an assessed quiz. Solutions to the problems and practice quizzes can be found in the instructor-provided lecture notes. Download the lecture notes from the link https://www.math.hkust.edu.hk/~machas/differential-equations-for-engineers.pdf Watch the promotional video from the link https://youtu.be/eSty7oo09ZI **Difficulty:** Beginner | **Language:** English | **Duration:** 6 weeks of study, 4 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/differential-equations-engineers **Tags:** courses, mathematics-statistics, calculus ## Books ### 1. The Laplace Transform: Theory and Applications **Author:** Joel L. Schiff Springer undergraduate text developing the transform rigorously: existence and basic principles, applications to ODEs, integral and difference equations, the Dirac delta via Riemann-Stieltjes integrals, a complex variable chapter, the complex inversion formula, and PDEs, with many worked examples. Next step after the computational toolkit: proves what the tables assume and covers the Bromwich inversion integral. More accessible and modern than Widder. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387986987?tag=edmonddante07-20 **Tags:** laplace-transform, inverse-laplace-transform, differential-equations, integral-equations, complex-analysis ### 2. A Student's Guide to Laplace Transforms **Author:** Daniel Fleisch Short conceptual text explaining what the transform does, its convergence region, and its relationship to Fourier and Z-transforms, then applying it to ODEs, PDEs, and engineering systems. Fills the gap between intuition videos and theory texts for self-learners who want the meaning of s and the region of convergence, not just a table. Author-produced podcasts and interactive solutions to every problem are free on the book's website. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/100909629X?tag=edmonddante07-20 **Tags:** laplace-transform, region-of-convergence, differential-equations, fourier-transform, z-transform ### 3. Notes on Diffy Qs, Chapter 6: The Laplace Transform **Author:** Jiří Lebl Free open textbook chapter covering the transform and its inverse, transforms of derivatives and solving ODEs, convolution, the Dirac delta and impulse response, and applying the transform to PDEs, with exercises and answers. A real textbook treatment with proofs-light but careful reasoning and graded exercises, between Paul's Notes (procedural) and Schiff (theory). Also available as a print paperback. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.jirka.org/diffyqs/html/laplace_section.html **Tags:** laplace-transform, differential-equations, convolution, dirac-delta, open-textbook ### 4. The Laplace Transform **Author:** David V. Widder Classic 1941 Princeton monograph, now a Dover reprint, treating the Laplace transform as rigorous real analysis: Stieltjes integrals, convergence, inversion and representation theorems, the moment problem, completely monotonic functions, Tauberian theorems, and the bilateral and Stieltjes transforms. Gives analysis-trained readers the transform's full theory. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/048647755X?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, differential-equations ## Websites ### 1. Paul's Online Notes: Laplace Transforms **Author:** Paul Dawkins Best free source of procedural fluency: partial-fraction inversions and step/delta IVPs worked line by line, which is where students actually get stuck. Ten-section chapter with worked examples: definition, computing transforms and inverses, step functions, solving IVPs, nonconstant-coefficient IVPs, discontinuous forcing, the Dirac delta, convolution integrals, and a printable transform table. Practice problems with solutions accompany each section. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx **Tags:** laplace-transform, inverse-laplace-transform, initial-value-problems, step-functions, differential-equations ### 2. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Youtubes ### 1. But what is a Laplace Transform? (3Blue1Brown Laplace series) **Author:** Grant Sanderson Animated 35-minute explanation of the integral definition, why exponentials are the building blocks, analytic continuation, and how poles of F(s) reveal a signal's exponential components. Part of a trilogy with a prelude on complex exponents and a follow-up on solving ODEs. Supplies the 'why' that table-driven courses skip: what s means, why poles matter, why differentiation becomes multiplication by s. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.youtube.com/watch?v=j0wJBEZdwLs **Tags:** laplace-transform, complex-exponentials, poles, differential-equations, visual-intuition --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/laplace-transforms