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If $$y_1=\sin(x^2)$$ $$y_2=\cos(x^2)$$ are linearly independent solutions of $$xy''-y'+4x^3y=0$$ the wronskian $$W(y_1,y_2)$$ is zero when $x=0$.

Does this contradict the theorem which states that if y1 and y2 are linearly independent solution then their wronskian is not zero ?

Are $y_1$ and $y_2$ still independent when x=0 but the wronskian is zero ?

Are $y_1$ and $y_2$ independent for all values of $x$ but dependent when $x=0$ ?

Is this related to the uniqueness theorem ? how ?

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The Wronskian of linearly independent functions may vanish at certain values of $x$ (or whatever variable the functions are in). However, the Wronskian will never be identically zero (i.e. it does not vanish for all values of $x$).

See this link.

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  • $\begingroup$ Since the Wronskian is a solution of Adel's equation, it is identically zero or never zero. $\endgroup$
    – Wang
    Oct 25, 2017 at 5:54

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