Does a Zero Wronskian Imply Linear Dependence for Smooth Functions?

[Bipolarity]

Suppose I have some functions $\{ y,y_{1},y_{2},...y_{n} \} \subset C^{∞}$ and suppose I know that the Wronskian of these functions is 0. Then can I conclude that these functions are linearly dependent?

Certainly this need not be true for an arbitrary set of functions, but it appears that it is true for analytic functions. My knowledge of these functions is very limited so I won't pursue it much, but are functions in $C^{∞}$ considered analytic?

Thanks for the clarification.

BiP

 

[Mandelbroth]

Suppose I have some functions $\{ y,y_{1},y_{2},...y_{n} \} \subset C^{∞}$ and suppose I know that the Wronskian of these functions is 0. Then can I conclude that these functions are linearly dependent?

Certainly this need not be true for an arbitrary set of functions, but it appears that it is true for analytic functions. My knowledge of these functions is very limited so I won't pursue it much, but are functions in $C^{∞}$ considered analytic?

Thanks for the clarification.

BiP

Functions in $C^{\infty}$ are considered smooth, but not necessarily analytic. See here for details.

If the Wronskian is 0, the functions are not necessarily linearly dependent. For example, consider $x^2$ and $x|x|$, the classical example given by Peano. Their Wronskian is 0, but they are clearly independent in any neighborhood of 0.