Separating Variables Visually

Differential Equations · Axiom Academy

Learn the intuition behind separation of variables — physically sorting a differential equation's terms so y lands on one side and x on the other. A separable differential equation like looks tangled — y and its rate of change sit mixed in with x . But if you can peel it apart so every y (and dy ) lands on one side and every x (and dx ) lands on the other, both sides become ordinary integrals you already know how to solve. Watch get pulled apart: the y terms slide to the left, the x terms slide to the right, until the equation reads — separated, ready to integrate. Once each side only has one variable, turns a differential equation into two ordinary integrals. Starting from , drag every term into the side it belongs on: all y -related terms (including dy ) on the left , all x -related terms (including dx ) on the right . Strategy: multiply both sides by y^2 to clear the denominator, then by dx to move it across — that's exactly what dragging the terms does here. An equation is separable exactly when it can be written as a PRODUCT . Click each card below to check whether it factors that way. Watch out: xy+x LOOKS like it has an addition, but it factors to x(y+1) — separable. A true non-separable equation like x^2+y^2 can't be pulled apart into a product no matter how you regroup it.

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