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:

$p_1(x,y,y^{\prime },\dots ,y^{(k-1)})y^{(k)}+p_2(x,y,y^{\prime },\dots ,y^{(k-1)})=0$

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:

$y^{(k)}=\frac{-p_2(x,y,y^{\prime },\dots ,y^{(k-1)})}{p_1(x,y,y^{\prime },\dots ,y^{(k-1)})}$

However, when doing this division, we may throw away some solutions where $p_1(x,y,y^{\prime },\dots ,y^{(k-1)})=p_2(x,y,y^{\prime },\dots ,y^{(k-1)})=0$. 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.