Linear First-Order Theory

Differential Equations · Axiom Academy

LESSON Linear First-Order Theory The standard form and the integrating factor — a method that always works, because it always turns the equation into a derivative you can just integrate. Every first-order linear differential equation can be written as: Here P(x) and Q(x) are given functions of x (constants count too). The equation is called linear because y and y' each appear only to the first power, with no products or compositions of them — no y^2 , no , no . 2. Building the Integrating Factor The move that makes linear equations always solvable: multiply both sides by a special function , the integrating factor , built directly from P(x) : This exact form isn't arbitrary. Integrate P(x) , then exponentiate — and that single design choice is what makes the left side of the equation collapse into a perfect derivative, which is the whole reason this method works (next step). 3. The Collapse — Why μ Makes It Work Watch what happens when we multiply the standard-form equation through by : The two terms on the left, and , are exactly expanded by the product rule — because we built so that . That's not a coincidence; it's the entire design of the integrating factor. 4. Integrate, Then Solve for y Once the left side is a single derivative, integrating both sides with respect to x is immediate: Divide by (always safe, since ) to isolate y :

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