Coupled Tanks

Differential Equations · Axiom Academy

Two tanks trade salt water back and forth — and suddenly one equation isn't enough. See why systems of differential equations exist. One tank is simple. Two connected tanks are not. A single tank with salt water flowing in and out is easy: its concentration depends only on itself. But connect a SECOND tank — with liquid flowing both ways between them — and each tank's rate of change now depends on the OTHER tank too. Watch both situations side by side. On the left, a lone tank settles by itself. On the right, two tanks trade salt water — Tank 1 feeds Tank 2, and Tank 2 feeds part of it back. Watch how the coupled pair's curves bend in response to EACH OTHER, not just themselves. The lone tank's curve only ever answers to itself. The coupled pair's curves visibly RESPOND to each other — Tank 2 even rises before it falls, fed by Tank 1's outflow. Adjust the flow between the tanks Pure water enters Tank 1 at a fixed rate. Drag the slider to change how fast liquid flows BACK from Tank 2 into Tank 1 — the forward flow (Tank 1 → Tank 2) and the drain automatically adjust to keep both tanks' volumes constant. Watch both concentration curves bend as you change the coupling. Every setting stays balanced automatically — the forward flow and drain rate both auto-derive from the back-flow you set, so the tank volumes never secretly grow or shrink. That balance is what makes this a genuine dX/dt = AX system. Every trajectory is a blend of two natural modes

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