 Solution

ID: 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:

W^{'} = u_{1} u_{2}^{''} - u_{2} u_{1}^{''} = u_{1} (- a u_{2}^{'} - b u_{2}) - u_{2} (- a u_{1}^{'} - b u_{1}) = - aW .

This is the constant-coefficient form of Abel's identity. Hence

W (x) = C e^{- a x}

, 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.

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