A common misconception in analysis is that if two $C^1$-functions are linearly independent then their Wronskian is non-zero at some point.
This is the result of carelessly reversing a true implication. The mistake was first made in 1882 by Thomas Muir (who is responsible for the name "Wronskian") in his "Treatise on the Theory of Determinants" and Giuseppe Peano pointed out a counterexample as early as 1889 (it suffices to take $y=x^2$ and $y=x|x|$).
For more information on the history of this mistake see "Peano on Wronskians".