Linear first-order equation

Definition

Let $I$ be an open interval and let $p, q : I \to \mathbb{R}$ be continuous functions. The linear first-order equation with coefficient $p$ and right side $q$ is the equation $$y' + p(x)\, y = q(x)$$ for an unknown differentiable function $y : I \to \mathbb{R}$. It is homogeneous if $q = 0$ on $I$, and inhomogeneous otherwise.

Remarks

The solution formula gives each solution explicitly, with an integrating factor. The homogeneous equation $y' + p(x) y = 0$ has the solutions $y = C e^{-P(x)}$, where $P$ is an antiderivative of $p$ and $C$ is a constant. Many authors write the equation as $y' = a(x) y + b(x)$, with $a = -p$ and $b = q$.

Explanations

L1 human

The equation is linear because $y$ and $y'$ appear only to the first power, with coefficients that depend on $x$. There is no product such as $y\, y'$ and no function such as $\sin y$. This structure is the reason why there is a formula for all the solutions.