First-Order Methods Summary

Differential Equations · Axiom Academy

Three techniques — separation, integrating factors, and exactness — all turn a first-order differential equation into a direct integration problem. Every first-order method works by transforming the equation into something you can integrate directly — solving is really just recognizing which transformation applies. Check the methods in a fixed order: separation first (fastest), then linear/integrating factor, then exactness — each covers equations the previous one can't handle. Watch for lost solutions: dividing by h(y) during separation can silently drop solutions where h(y) = 0 . The same three equation types model real systems: separable equations give growth/decay and cooling; linear equations give mixing tanks and RC circuits. An initial condition pins down the arbitrary constant C that every general solution carries. Method 1 Separation of Variables Use whenever the right-hand side factors into a function of x alone times a function of y alone. Move every y -term to one side, every x -term to the other, then integrate both sides independently. When to use: the equation splits into g(x)h(y) — this is the first thing to check, since it's the fastest method. Watch out for: dividing by h(y) can lose constant solutions where h(y) = 0 — check those separately. Method 2 Linear / Integrating Factor

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