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Is this true?

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

No — it is false.

The counterexample

|x| on

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

The kind of mistake this is

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this
ODEwronskian

Others that fail the same way

From Ordinary Differential Equations › Linear ODE, Wronskian, variation of parameters, systems

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