From ODEs to PDEs

Differential Equations · Axiom Academy

Everything so far has changed with just one variable — time. Now watch what happens the moment a second variable, space, joins in. One clock becomes a whole coordinate Every differential equation you've solved so far has tracked a single quantity as ONE variable — time — pushed it forward. That's an ordinary differential equation (ODE): u depends only on t . The instant a quantity also depends on where you are, not just when , the derivatives split into pieces — and you're doing a partial differential equation (PDE) instead. Watch the cup cool first — one number, one clock. Then watch the rod: the SAME idea (heat leaving a hot object) now needs a whole temperature profile along its length, reshaping as time runs. Same physics (Newton's cooling), one extra independent variable — and the equation itself changes character. Two directions to differentiate in Drag along the rod (frozen at one instant). At each point you can ask two different questions — "how fast is it heating up right now ?" ( ) and "how steeply does it change along the rod ?" ( ) — and they need different derivatives to answer. Notice: and always match here — that agreement is the heat equation, u_t = u_ xx , holding at every point. Click through the equations below. The heat equation is only the first PDE you'll meet — the same "more variables → partial derivatives" idea produces the wave equation and Laplace's equation too.

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