--- title: "Direction Fields and Euler's Method" description: "Draw the slope the equation prescribes at each point and the solution curves appear without any formula. Stepping along those slopes numerically is Euler's method, with error you can reason about." canonical: https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR source: https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Direction Fields and Euler's Method Draw the slope the equation prescribes at each point and the solution curves appear without any formula. Stepping along those slopes numerically is Euler's method, with error you can reason about. Subject: Mathematics · Area: Differential Equations · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/direction-fields-and-eulers-method-mt_Ak-TZOX3GR ## Ready when they can - Sketch solution curves on a given direction field - Take several Euler steps by hand and tabulate the result - Say how halving the step size affects the error, and why the method drifts ## Lesson: Slope maps and walking the field A direction field turns an equation into a picture. At each point you draw a tiny segment with the slope the equation prescribes there. A solution is a curve that stays tangent to every segment it crosses. You can read long term behaviour, like rest points and stability, straight from the picture. For y prime = x minus y, the slope at (0, 1) is 0 minus 1, which is minus 1. **Example.** Euler's method walks the field one short step at a time. From your point you follow the local slope for one step, land, re-measure, and repeat. The update is y next = y + h times slope. Take y prime = x + 1 with y(0) = 0 and step 1. At (0, 0) the slope is 0 + 1 = 1, so y1 = 0 + 1 times 1 = 1. Re-measure at (1, 1): the slope is 1 + 1 = 2, so y2 = 1 + 1 times 2 = 3. Your table reads (0, 0), (1, 1), (2, 3). Smaller steps track the true curve more closely, but you need more of them. The error shrinks roughly in proportion to the step size, so halving the step roughly halves the error. Euler drifts because it freezes the old slope across the whole step while the true slope keeps changing. It never follows the exact curve. **Tip.** To sketch a solution, draw a few segments first, then thread your curve through the starting point. Always evaluate the slope at the point you stand on, never the one you are heading to. When a table drifts, check for a miscopied step size or a slope taken at the wrong point. **Recap.** Read slopes from the field, step with the local slope, and shrink the step to cut the drift. ## Practice 13 questions on this page, each with its working shown. ## Needs first - [The Newton-Raphson Method](https://lightmysky.com/learn/mathematics/the-newton-raphson-method-mt_jkjJre5ETd) - [What a Differential Equation Says and What a Solution Is](https://lightmysky.com/learn/mathematics/what-a-differential-equation-says-and-what-a-solution-is-mt_tHlHg2nK3o) ## Opens up - [The Wave Equation and Its Characteristics](https://lightmysky.com/learn/mathematics/the-wave-equation-and-its-characteristics-mt_-_QCiBWg1w) - [Separable Equations](https://lightmysky.com/learn/mathematics/separable-equations-mt_itSTsLTkwQ) - [Numerical Schemes for Evolution Equations](https://lightmysky.com/learn/mathematics/numerical-schemes-for-evolution-equations-mt_RuDbOapexB) - [Phase Portraits, Equilibria and Stability](https://lightmysky.com/learn/mathematics/phase-portraits-equilibria-and-stability-mt_ULuZk4lmGr)