Is Wronskian infinitely times differentiable?

Question

If $y_1, y_2$ are two solutions of the differential equation

$a_{0}(x)y'' + a_{1}(x)y' +a_{2}(x)y = 0$ where $a_{0},a_{1} a_{2}$ are continuous and $a_{0}(x) \ne 0$

Then the wronskian $W$ of $y_1,y_2$ is infinitely times differentiable (True/False)

Now, this is my approach using Abel's Identity this is equal to

$W = A\text{exp}(-\displaystyle\int \dfrac{a_1(x)}{a_{0}(x)} dx)$

So, by this logic Wronskian seems to be infinite times differentiable .

Is this correct ?

Can someone please verify ?

Thank you.

Answer 1

False. If the $a_i$ are only continuous, there is no reason for $W$ to be differentiable more than once.