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.
What a learner can do afterwards
- 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
1 · Read
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.
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.
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.
Read slopes from the field, step with the local slope, and shrink the step to cut the drift.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.