Homogeneous linear equation

Definition

Let $I$ be an open interval and let $a_0, \dots, a_{n-1} : I \to \mathbb{R}$ be continuous functions. A homogeneous linear equation of order $n$ is a linear differential equation with right side $0$: $$y^{(n)} + a_{n-1}(x)\, y^{(n-1)} + \dots + a_1(x)\, y' + a_0(x)\, y = 0.$$ For a linear equation $L y = b$, the equation $L y = 0$ is the associated homogeneous equation.

Remarks

The zero function is always a solution. By the superposition principle, the solutions on $I$ form a vector space. It has dimension n. Example: $y'' - y = 0$ is homogeneous, and its solutions are $C_1 e^{x} + C_2 e^{-x}$.

Explanations

L1 human

Homogeneous means that the equation has no term without $y$. So if $y$ is a solution, then $2y$ and $-y$ are solutions, and so is the sum of two solutions.