Linear equation with constant coefficients

Definition

Let $n \ge 1$, let $a_0, \dots, a_{n-1}$ be real numbers, let $I$ be an open interval and let $b : I \to \mathbb{R}$ be continuous. A linear equation with constant coefficients is a linear differential equation of the form $$y^{(n)} + a_{n-1}\, y^{(n-1)} + \dots + a_1\, y' + a_0\, y = b(x).$$ Its homogeneous equation is the case $b = 0$, on the interval $I = \mathbb{R}$.

Remarks

The function $e^{r x}$ solves the homogeneous equation if and only if $r$ is a root of the characteristic equation $r^n + a_{n-1} r^{n-1} + \dots + a_0 = 0$. The roots of this polynomial give a fundamental system. See the cases of distinct real roots, repeated roots and complex conjugate roots. The harmonic oscillator $x'' + \omega^2 x = 0$ is an example of order $2$.

Explanations

L1 human

When the coefficients are constants, the derivative of $e^{r x}$ is $r e^{r x}$, again a multiple of the same function. So the equation becomes a polynomial equation in $r$. This changes calculus into algebra.