Splitting and Integrating

Calculus 2 · Axiom Academy

A first-order equation that tangles x and y together can sometimes be pulled apart — each variable to its own side — and then integrated piece by piece. One trick unlocks a whole family of equations A differential equation like mixes x and y together, so you can't just integrate it. But notice it's a product: something in x times something in y . When that's true, the variables can be separated — herded onto opposite sides of the equals sign — and once apart, each side integrates on its own. Watch the equation reorganize itself. Treating as a ratio of differentials, every y -piece drifts left and every x -piece drifts right, until the tangled equation has become two clean, single-variable sides. The same equals sign, the same equation — just sorted so each side speaks only one variable. You decide which side each piece belongs on Here's a different equation, . Dividing by and multiplying by leaves four pieces. Tap each one to send it to its side — anything carrying y goes left, anything carrying x goes right. Sorted, the four pieces read — every y on the left, every x on the right. Two single-variable sides, two independent integrals Back to the first equation, now separated as . Slide the handle to sweep the integral sign across both sides at once. Each side resolves on its own — the left in y , the right in x — and a single constant ties them together. One differential equation, traded for two integrals you already know how to do. The whole method, in three moves

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