Past exam of the mathematical methods I course of the University of Oxford / 2019 resit / 12 / b / Solution

Differentiate the Wronskian and cancel the mixed first-derivative terms:This is the constant-coefficient form of Abel's identity. Hence $W(x)=C{e}^{-ax}$, or $W(x)=W(x_0)e^{-a(x-x_0)}$. Since the exponential function is never zero, $W$ is either identically zero or nowhere zero on a connected interval of the homogeneous equation.