Exact Equation Theory

Differential Equations · Axiom Academy

A first-order equation is exact when it's secretly the total differential of a hidden function — find that function, and the solution falls out for free. 1. What Makes an Equation "Exact"? A first-order equation is called exact if there's a function F(x,y) whose total differential matches it term for term: The total differential of any F(x,y) Exactness: matching M to F_x and N to F_y Once F is known, every solution curve is a level curve of that potential — a slice where F holds a constant value. That's why the general solution is just F(x,y)=C : watch the surface get built below, one level circle at a time. We rarely get handed F directly — we need a test we can run on M and N alone. It comes from a fact about mixed partial derivatives : if F is smooth enough, differentiating in either order gives the same result. Test it on a pair below: M=2xy , N=x^2 . Compute and side by side and see whether they land on the same value. Once exactness is confirmed, we rebuild F from its own partial derivatives — a four-step pipeline of partial integration and matching: Integrate M(x,y) with respect to x (treat y as constant). Add an unknown function g(y) — anything that vanishes under . Take of that result and set it equal to N(x,y) . Match terms to isolate g'(y) , then integrate once more to get g(y) . Assemble F(x,y) and the implicit general solution is F(x,y)=C . Put the pipeline to work on a real equation: Since , the equation is exact. Integrate M with respect to x:

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