Mr. Integral
Math notes/Differential Equations/Equations and Initial Values

Direction Fields and Solution Curves — Cheat sheet

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Core rule

At (t,y)(t,y), the tangent slope is f(t,y)f(t,y). Zero-slope curves need not themselves be solutions.

Watch for

Under uniqueness, distinct solution curves cannot cross at the same (t,y)(t,y). A coarse slope field can hide rapid changes; numerical traces need a step-size check as well as a visually plausible curve.