Explicit differential equation

Definition

An explicit differential equation is a differential equation where the highest-order derivative is explicitly written as a function of the independent variable, dependent variable, and lower order derivatives. Explicitly, if the independent variable is $x$, the dependent variable is $y$, and the order is $k$, it has the form:

$y^{(k)}=G(x,y,y',\dots ,y^{(k-1)})$

where $G$ is a function of $k+1$ variables.

Relation with first-degree differential equations

Any explicit differential equation is a first-degree differential equation. Conversely, any first-degree differential equation can be converted to an explicit differential equation via division by a function, though we may need to add back certain solutions that are identically zeros for the function.