  Given an ordinary differential field $K$ of characteristic zero, it is known
that if $y$ and $1/y$ satisfy linear differential equations with coefficients
in $K$, then $y'/y$ is algebraic over $K$. We present a new short proof of this
fact using Gr\"{o}bner basis techniques and give a direct method for finding a
polynomial over $K$ that $y'/y$ satisfies. Moreover, we provide explicit degree
bounds and extend the result to fields with positive characteristic. Finally,
we give an application of our method to a class of nonlinear differential
equations.
