{wiki}

The name is a bit obscure if you don't think in very generalized terms right out of the gate. It refers to a <linear polynomial> of <multivariate polynomial>[multiple variables], which by definition must have the super simple form of:
$$
f(x_0, x_1, ..., x_n) = c_0x_0 + c_1x_1 + ... + c_nx_n + k
$$
and then we just put the unknown $y$ and each derivative into that simple polynomial:
$$
f(y(x), y'(x), ..., y^{(n)}(x)) = c_0y + c_1y' + ... + c_ny^{(n)} + k
$$
except that now the $c_i$ are not just constants, but they can also depend on the argument $x$ (but not on $y$ or its derivatives).

Explicit solutions exist for the very specific cases of:
* constant coefficients, any degree. These were known for a long time, and are were studied when <Ciro Santilli's formal education>[Ciro was at university] in the <University of São Paulo>.
* degree 1 and any coefficient


 Linear differential equation