First-degree differential equation

Definition

A first-degree differential equation is a differential equation that is linear in its highest-order derivative. Explicitly, of the independent variable is $x$, the dependent variable is $y$, and the order is $k$it has the form:

$p_{1}(x,y,y',\dots ,y^{(k-1)})y^{(k)}+p_{2}(x,y,y',\dots ,y^{(k-1)})=0$

Relation with explicit differential equations

First-degree differential equations are typically solved by converting them to explicit differential equations. This simply involves dividing by $p_{1}$ and rearranging. The explicit differential equation form is:

$y^{(k)}={\frac {-p_{2}(x,y,y',\dots ,y^{(k-1)})}{p_{1}(x,y,y',\dots ,y^{(k-1)})}}$

However, when doing this division, we may throw away some solutions where $p_{1}(x,y,y',\dots ,y^{(k-1)})=p_{2}(x,y,y',\dots ,y^{(k-1)})=0$. These solutions can be checked for separately, but we can note that these are differential equations of a lower order (specifically, order at most $k-1$) hence presumably easier to solve.