The Integrating Factor Magic

Differential Equations · Axiom Academy

Watch two stubborn terms fuse into a single derivative — and see exactly why multiplying by the right function makes any linear first-order equation solvable. A left side that refuses to integrate Take the equation y' + 2xy = x . You can't just integrate both sides — the left side is two separate terms, y' and 2xy , that don't combine into anything you recognize as a derivative. But multiply the entire equation by one specially-chosen function, , and something remarkable happens: those two terms stop being separate. They fuse into a single derivative, . Watch it happen. The two terms on the left, and , slide together and lock into one combined expression — because was built so that , exactly the condition the product rule needs. The product rule says — so once holds, the messy left side of the ODE IS that derivative, in disguise. The multiplier comes straight from P(x) For an equation y' + P(x)y = Q(x) , the integrating factor is always — built entirely from P , the coefficient sitting on y . Drag the slider to change P(x) = kx and watch update to match, live. Every you draw satisfies — that's the ONE property that makes the collapse from Component 0 work, no matter what P is. Integrate once, divide by , done Once the left side collapsed to , the equation integrates directly: , so . Drag C to explore the whole family of solutions — every curve you draw solves the original equation y' + 2xy = x .

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