Abel's identity

Statement

Let $I$ be an open interval, let $p, q : I \to \mathbb{R}$ be continuous, and let $y_1, y_2$ be solutions on $I$ of $$y'' + p(x)\, y' + q(x)\, y = 0.$$ Let $W = y_1 y_2' - y_1' y_2$ be their Wronskian, and let $x_0 \in I$. Then for each $x \in I$, $$W(x) = W(x_0) \exp\left( -\int_{x_0}^{x} p(t)\, dt \right).$$

Remarks

Proof idea: a computation with the equation gives $W' = -p W$, a linear first-order equation for $W$. So $W$ is zero at each point or at no point of $I$. For an equation of order $n$ with coefficient $a_{n-1}$ of $y^{(n-1)}$, the same formula holds with $p$ replaced by $a_{n-1}$. Liouville's formula extends it to linear systems. Niels Henrik Abel found the identity.

Explanations

L1 human

You can find the Wronskian of two solutions without knowing the solutions. Only the coefficient $p$ of $y'$ matters. For $y'' + y = 0$, we have $p = 0$, so the Wronskian of two solutions is constant.