Number of zeros of Wronskian

Question

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?

Answer 1

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.