Initial Conditions and Uniqueness

Differential Equations · Axiom Academy

LESSON Initial Conditions and Uniqueness How one starting point pins down a single solution curve — and the exact condition that guarantees it's the only one. 1. A Differential Equation Has a Whole Family of Solutions Take the simplest possible growth equation: Every function of the form below solves it — one curve for every real number C : Change C and you get a different solution curve, but the differential equation itself never picks a favorite — it's satisfied by the whole family at once. 2. An Initial Condition Selects Exactly One Curve Geometrically, the condition y(t_0) = y_0 drops a pin at the point (t_0, y_0) . Out of every curve in the family, only one passes through that exact pin — and that curve is the solution to the IVP. For y(0) = 2 , only C = 2 makes the curve pass through that point, so the IVP's unique solution is: 3. When Is That Unique Curve Guaranteed to Exist? Existence: a solution exists on some interval around t_0 Uniqueness: that solution is the only one through (t_0, y_0) The Lipschitz condition says f can't change too abruptly with respect to y — it bounds how fast nearby solution curves are allowed to pull apart: 4. When the Guarantee Breaks: Two Curves, One Point The theorem is a sufficient condition, not automatic — drop it and uniqueness can genuinely fail. Consider: with y(0) = 0 . This innocent-looking IVP has two different solutions that both start at the origin:

This is the written version of the interactive lesson above. See the full Differential Equations course.