# Number of zeros of Wronskian

Is there some relation between the number of zeros of a Wronskian and properties of given functions? Having Wronskian (e.g. $2$ x $2$) $$W(x)=\left|\begin{array}{c}f_1(x) & f_2(x)\\f'_1(x) & f'_2(x)\end{array}\right|$$

For example, the number of zeros, i.e., $W(x)=0$ is at most the number of zeros of the functions $f_1(x), f_2(x), \dots$, or anything similar.

I am trying to find any bound on number of zeros of any general Wronskian based on the functions. I am not interested in any sharp bound. Is there a result in literature regarding this problem?

• @Amzoti I mean the second case, e.g. for 2x2 case $W(x)=f_1(x)f_2'(x)-f_2(x)f_1'(x)=0$ and I have some information about $f_1(x), f_2(x)$, and possibly about their derivatives (e.g. number of roots). And then I want to learn about the number of roots of $W(x)=0$ based on this information. Commented Mar 27, 2014 at 11:47

I don't think, you can conclude anything. Let us take a few examples.

Example $$1$$: Take $$f_{1}(x) = x+1$$ and $$f_{2}(x) = x^{2}+1$$

Then Wronskian, $$W(x)=\begin{vmatrix} f_{1}(x) & f_{2}(x) \\ f'_{1}(x) & f'_{2}(x) \end{vmatrix} =\begin{vmatrix} x+1 & x^{2}+1 \\ 1 & 2 x \end{vmatrix} =x^{2}+2 x-1$$

Now $$W(x)=0 \implies x^{2}+2 x-1=0 \implies x= -1 \pm \sqrt{2}$$

Clearly, there is no similarity of roots of $$W(x)=0$$ with $$f_{1}(x)=0$$ and $$f_{2}(x)=0$$. Now if you conclude that the number of roots of $$W(x)=0$$ is lcm (least common multiple) of the number of roots of $$f_{1}(x)$$ and $$f_{2}(x)$$, then take a look on the next example.

Example $$2$$: Take $$f_{1}(x) = x+1$$ and $$f_{2}(x) = x-1$$

Then Wronskian, $$W(x) =\begin{vmatrix} x+1 & x-1 \\ 1 & 1 \end{vmatrix} = 2$$.

In this case if you make $$W(x) =0$$, then it a inconsistent and therefore how can you apply your previous finding to conclude this type of problem ?

Example $$3$$: Take $$f_{1}(x) = \sin x$$ and $$f_{2}(x) = 2 \sin x$$

Clearly, these two functions are linearly dependent and so here $$W(x) =0$$, again you can't apply your previous finding to conclude any thing.