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 multiple variables, which by definition must have the super simple form of:
and then we just put the unknown $y$ and each derivative into that simple polynomial:
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).

$f(x_{0},x_{1},...,x_{n})=c_{0}x_{0}+c_{1}x_{1}+...+c_{n}x_{n}+k$

$f(y(x),y_{′}(x),...,y_{(n)}(x))=c_{0}y+c_{1}y_{′}+...+c_{n}y_{(n)}+k$

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 was at university in the University of São Paulo.
- degree 1 and any coefficient