--- title: Initial Value Problems description: Initial value problems pair a differential equation with specified values of the unknown function at a single point. Learners will understand existence and uniqueness theorems and solve equations modeling time-dependent processes. category: mathematics subcategory: differential-equations difficulty: beginner, intermediate, advanced url: /subject/initial-value-problems --- # Initial Value Problems Initial value problems pair a differential equation with specified values of the unknown function at a single point. Learners will understand existence and uniqueness theorems and solve equations modeling time-dependent processes. ## Available Resources 1 Videos • 3 Books • 3 Courses • 2 Websites ## Websites ### 1. Paul's Online Math Notes - Differential Equations Paul Dawkins' free Lamar University notes for a first ODE course, with worked examples on first-order equations, second-order linear equations, Laplace transforms, systems and series solutions, so learners can solve initial value problems by hand step by step. **Difficulty:** Beginner | **Price:** Free **Link:** https://tutorial.math.lamar.edu/Classes/DE/DE.aspx **Tags:** differential-equations, ordinary-differential-equations, initial-value-problems, laplace-transforms ### 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 ## 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. Qualitative Behavior: Phase Portraits (MIT 18.03SC) **Author:** Arthur Mattuck, Haynes Miller, Jeremy Orloff et al. Covers sketching trajectories of two-dimensional linear systems in the phase plane, classifying critical points as nodes, saddles, spirals or centers, and reading stability from the trace-determinant diagram. Includes notes, problem-solving videos, Mathlet demonstrations and practice problems with solutions, so learners can sketch portraits unaided. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/pages/unit-iv-first-order-systems/qualitative-behavior-phase-portraits/ **Tags:** phase-portraits, linear-systems, critical-points, stability-analysis, differential-equations ### 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. Differential Equations, Dynamical Systems, and an Introduction to Chaos (3rd Edition) **Author:** Morris W. Hirsch, Stephen Smale, Robert L. Devaney Proof-based successor to Hirsch and Smale's classic, building phase space theory from linear systems and canonical forms through nonlinear flows, equilibria and stability, limit sets, the Poincare-Bendixson theorem, bifurcations, and chaotic attractors, with exercises. **Difficulty:** Advanced | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0123820103?tag=edmonddante07-20 **Tags:** differential-equations, dynamical-systems, phase-portraits, bifurcation-theory, chaos-theory ### 2. Exploring ODEs **Author:** Lloyd N. Trefethen, Ásgeir Birkisson, Tobin A. Driscoll SIAM textbook by Trefethen, Birkisson and Driscoll, released free as a PDF by the authors. Treats initial value problems, boundary value problems, stiffness, blow-up and chaos, illustrating each concept with short Chebfun computations and roughly 400 generated figures. Bridges the gap between symbolic IVP technique and what solutions really do: every example is solved numerically and plotted, so learners see stiffness, sensitivity to initial data and finite-time blow-up rather than reading about them. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://people.maths.ox.ac.uk/trefethen/ExplODE/ **Tags:** ordinary-differential-equations, numerical-methods, chebfun, dynamical-systems, stiff-equations ### 3. Ordinary Differential Equations and Dynamical Systems **Author:** Gerald Teschl Graduate textbook published as AMS Graduate Studies in Mathematics 140, with the full PDF free on the author's university page. Part I develops initial value problems rigorously: Picard-Lindelöf existence and uniqueness, dependence on initial conditions, and extensibility. **Difficulty:** Advanced | **Language:** English | **Price:** Free **Link:** https://www.mat.univie.ac.at/~gerald/ftp/book-ode/ **Tags:** ordinary-differential-equations, dynamical-systems, existence-and-uniqueness, picard-lindelof, chaos-theory ## Videos ### 1. Differential Equations, Studying the Unsolvable (3Blue1Brown) **Author:** Grant Sanderson Opening chapter of Grant Sanderson's differential equations series. Uses the pendulum to show why most differential equations have no closed-form solution, introducing state vectors, phase space, and step-by-step numerical integration from an initial condition. Gives the geometric picture, a point moving through phase space under a vector field, that makes the initial condition feel like a starting position rather than an arbitrary constant. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.3blue1brown.com/lessons/differential-equations **Tags:** differential-equations, initial-value-problems, phase-space, numerical-integration, pendulum --- *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/initial-value-problems