Fundamental system of solutions

Definition

Let $I$ be an open interval and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous. Let $S$ be the vector space of solutions on $I$ of the homogeneous linear equation $y^{(n)} + a_{n-1}(x) y^{(n-1)} + \dots + a_0(x) y = 0$. A fundamental system of solutions of the equation is a basis $(y_1, \dots, y_n)$ of $S$.

Remarks

Since $S$ has dimension n, a family of $n$ solutions is a fundamental system if and only if it is linearly independent. The Wronskian test checks this. Each solution is then $c_1 y_1 + \dots + c_n y_n$ for unique constants $c_1, \dots, c_n$. Example: $(\cos x, \sin x)$ is a fundamental system of $y'' + y = 0$ on $\mathbb{R}$.

Explanations

L1 human

A fundamental system is a set of building blocks. Each solution of the homogeneous equation is a unique combination of these $n$ solutions.