10 Linear differential operators

Let $t$ be the independent variable. By combining (i) and (iii), we can construct linear differential operators

$$Lf(t)=a_n(t)\frac{d^{n}f}{d{t}^n}+a_{n-1}(t)\frac{d^{n-1}f}{d{t}^{n-1}}+\mathrm{\dots }+a_0(t)f(t);$$

the number of derivatives $n$ is the order of $L$; the $a_j(t)$ are the coefficient (functions). When the $a_j$ are constants, we talk about a linear differential operator with constant coefficients. A linear equation of order $n$ is

$$a_n(t)\frac{d^{n}f}{d{t}^n}+a_{n-1}(t)\frac{d^{n-1}f}{d{t}^{n-1}}+\mathrm{\dots }+a_0(t)f(t)=u(t),$$

where $a_n(t),\mathrm{\dots },a_0(t),u(t)$ are given and $f(t)$ is to be found.