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Counterexample

Is this true?

W(f, g) ≡ 0 implies f and g are linearly dependent

No — it is false.

The counterexample

f(x)=x2,g(x)=xf(x) = x^{2}, g(x) = x|x| on R\mathbb{R}

Why it fails

The Wronskian vanishes identically but no constant multiple relates them. The implication holds only for solutions of a common linear ODE.

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Standard counterexample

There is a canonical object that settles this. Recognising it is the whole question.

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Go deeper· Explained in full in Linear ODE, Wronskian and systems· Asked in CSIR NET June 2025· From Ordinary Differential Equations › Linear ODE, Wronskian, variation of parameters, systems

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