The Separation Method

Differential Equations · Axiom Academy

Sort a differential equation into a y-side and an x-side, integrate each independently, and the solution falls out. 1. Recognizing a Separable Equation A first-order equation is separable if it can be written as a product of a function of y alone and a function of x alone — never a sum, and never one variable trapped inside a function of the other. The separable form: a pure product Same equation, variables sorted onto their own sides 2. Integrate Both Sides — Worked in Full Once separated, slap an integral sign on each side and integrate independently. Every indefinite integral needs a constant, but since both sides get one, they combine into a single constant C on whichever side is more convenient. Drop it and every initial condition becomes unsolvable — C is what an IC pins down next. 3. Solve for y, Then Pin Down the Constant Undo the logarithm to get an explicit family of solutions, then use the initial condition to select the one curve that actually fits — the constant A is not free once a starting point is given. Exponentiating gives y = Ae^ x^2/2 for the constant . Setting x=0 forces A=1 , so the specific solution is the single curve shown settling onto the family above — every other member of the family is ruled out by the initial condition. 4. When Separation Fails — and What It Can Hide Not every first-order equation is separable, and even when it is, the algebra has a blind spot worth knowing about before it costs you a solution.

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