Linear differential equation of order n

Remarks

The map $L$ is linear: $L(c_1 y_1 + c_2 y_2) = c_1 L y_1 + c_2 L y_2$ for constants $c_1, c_2$. An equation $a_n(x) y^{(n)} + \dots + a_0(x) y = b(x)$ with continuous $a_n$ and no zero of $a_n$ on $I$ takes this form after division by $a_n$. The equation is homogeneous if $b = 0$. The case $n = 1$ is the linear first-order equation. For each $x_0 \in I$ and each $(c_0, \dots, c_{n-1}) \in \mathbb{R}^n$, there is exactly one solution on all of $I$ with $y^{(k)}(x_0) = c_k$ for $k = 0, \dots, n-1$.