If the Wronskian of two functions is zero then these functions are LD I'm studying a book of differential equations which says that if the Wronskian of two functions is zero then these functions are linearly dependent. the author doesn't prove it, he simply said as a easy consequence of basic properties of determinants, I tried to prove by myself without success. 
I need help.
thanks a lot.
 A: The statement holds if $f_1$ and $f_2$ are solutions of a linear differential equation. The functions $x^3$ and $|x|^3$ are linearly independent in any open interval containing $0$, but their Wronskian is identically equal to $0$.
A: This appears to be false: that is, you wan have the Wronskian of two functions being identically zero on an interval without those two functions being linearly dependent.
See for instance the Wikipedia article:

A common misconception is that $W = 0$ everywhere implies linear dependence, but Peano (1889) pointed out that the functions $x^2$ and $|x|x$ have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of 0. 

However, it holds with additional assumptions on the functions (eg, one of them never takes value $0$ — cf. same link).
A: Let $f_1, f_2$ be two differentiable functions.  The Wronskian of these functions is given by: $$f_1 f_2^\prime - f_2 f_1^\prime$$ and we are told that this difference is $0$.
This tells us that $$\frac{f_2^\prime}{f_2} = \frac{f_1^\prime}{f_1}$$ and so by integration we have $\ln(|f_1|) = \ln(|f_2|) + C$ for some constant $C$.  This yields $$f_1 = \pm e^C \cdot f_2.$$
Hence these functions are linearly dependent.
