Everything With a y on One Side

You'll learnto decide whether an equation separates before you start rearranging it, to bring the constant of integration in at the one step where it belongs, and to read off what that constant becomes once you solve for y.

Separation of variables is Unit 6's integration done twice at once, and almost all of its difficulty is bookkeeping. Move every y onto the side carrying dy and every x onto the side carrying dx, integrate both sides, then solve for y. Two things decide whether the answer comes out right. The equation has to separate at all, and that is a factoring question rather than a rearranging one — a sum like x + y does not factor, and no amount of dividing will make it. And the constant of integration enters at exactly one moment, the integration itself, so that exponentiating turns it into a factor instead of leaving it as a term. A constant multiplying an exponential and a constant added to one are different families of curves: the first stretches a single curve, the second slides it.

8 STEPS · 6 QUICK CHECKS · ONE CONSTANT, AND IT ENTERS AT THE INTEGRATION · v1

dy/dx = xy ⇒ dy/y = x dx ⇒ ln|y| = x²/2 + C ⇒ the constant becomes a factor

Before you start

What you're looking at

A derivation worked down the page, one line at a time. The constant of integration is drawn in teal wherever it stands alone, so you can watch which side it is on and what it turns into.

The question

Does this equation separate, and once both sides are integrated, where does the single constant of integration end up in the final answer?

Watch for

The step with two panels of curves. Both families use the same four constants, and they are not two drawings of one thing — one family stretches a curve and the other slides it.

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