First-order ordinary differential equation ($y^{\prime }=F(x,y)$)

A first-order ordinary differential equation contains the first derivative of its unknown function and no higher derivative. When solved for that derivative it has the displayed form. With an initial condition, local existence and uniqueness follow under appropriate regularity assumptions on $F$; the order alone does not guarantee either property.

On an interval with $x\ne 0$, the substitution $y=xv$ and the product rule give the displayed identity. An equation $x{y}^{\prime }-y=g(y/x)$ becomes $v^{\prime }=g(v)/x^2$ and permits separation of variables wherever $g(v)\ne 0$. Roots $g(v_0)=0$ also give solutions $y=v_{0}x$ and must be retained if division by $g$ was used. This substitution works for this displayed class even though its explicit factor $1/x$ means that $y^{\prime }=y/x+g(y/x)/x$ need not be a homogeneous equation of the usual form $y^{\prime }=F(y/x)$.