Linear First-Order Equations: Variation of the Constant
Mastering the two faces of the same powerful method — and the structure behind every linear first-order ODE
1. The Class Defined — What Makes an ODE "Linear"?
Standard Form
A first-order ordinary differential equation (ODE) is linear if it can be written in the standard form:
Here the word "linear" refers to how the unknown $y$ and its derivative $y'$ appear: each shows up at most to the first power, never multiplied together, never inside a nonlinear function like $\sin(y)$ or $e^y$. The coefficient $P(x)$ and the right-hand side (the forcing) $Q(x)$ may be any functions of the independent variable $x$.
- $P(x)$ — the coefficient of the unknown.
- $Q(x)$ — the forcing term (or right-hand side). It drives the equation away from zero.
- When we reach second-order linear equations later, the same vocabulary will reappear with an extra derivative term — everything you learn here transfers directly.
Normalising the Leading Coefficient
If an equation arrives in the form
$$ a_1(x)\,y' + a_0(x)\,y = g(x), $$we divide every term by $a_1(x)$ (assuming $a_1(x)\neq 0$) to reach the standard form with the derivative's coefficient equal to 1:
$$ y' + \underbrace{\frac{a_0(x)}{a_1(x)}}_{P(x)}\,y = \underbrace{\frac{g(x)}{a_1(x)}}_{Q(x)}. $$⚠ Classic entry error: Forgetting to normalise before identifying $P(x)$ is the single most common mistake with the integrating factor method.
Homogeneous vs. Non-Homogeneous
Homogeneous ($Q(x)=0$)
$$ y' + P(x)\,y = 0 $$No external forcing — the equation is "self-contained."