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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 Grobner 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.

Logarithmic derivatives of solutions to linear differential equations