Product Solutions

Differential Equations · Axiom Academy

Discover the single bold guess that turns a hard PDE into two easy ODEs: assume . One wave, two independent behaviors A vibrating string's height u(x,t) depends on where you look ( x ) and when you look ( t ) at once — that's what makes the wave equation u_ tt = c^2 u_ xx a genuinely hard partial differential equation. The single idea that unlocks it: guess that space and time act independently, so the solution is just a product of a shape and an envelope. Watch the wave pull apart into its two factors: a fixed spatial shape X(x) that never changes, and a temporal envelope T(t) that stretches it in and out — then watch them multiply back together into the exact same wave. The shape X(x) is frozen — only the envelope T(t) pulses it. That's the whole content of . Substitute, then divide — and the variables fall apart Plug the guess u(x,t)=X(x)T(t) into the wave equation and divide through by X(x)T(t) . Follow the four moves: Setting both sides to gives (spatial) and (temporal) — connected only by . Drag time — watch the shape hold still Slide t and watch three curves live: X(x) (always the same shape), T(t) (the envelope value at this instant), and their product u(x,t)=X(x)T(t) . The shape never redraws — only how far it's stretched changes. never changes; only the scalar stretches it toward zero and back — that's separability in motion. One drives both equations at once

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