The following theorem will be established:

Provided the roots of the associated characteristic equation are all real, any solution of a linear homogeneous differential equation with constant coefficients has at most n − 1 zeros for real finite values of the independent variable, where n is the order of the equation.

The theorem applies to equations with a complex independent variable, but since the conclusion concerns only real values of the variable there is no loss of generality in considering an equation of order n in the form

with y = y(x), where x is real. The constant coefficients ar may be complex.