I am writing my bachelor thesis to prove that a certain linear fractional differential equation of order $(n,q)$ has $N$ linearly independent solutions, where $N$ is the smallest possible integer greater than $nv$, where $v=1/q$. I understand the proof more or less, but now I desperately try to find an example. I have given an example which at first sight seemed relatively simple, but I have no idea how to come up with the solution.

So the example is

$$D^{4v}y(t)=0, ~~\text{with}~~ \ v=1/3 $$

So, if I now take the Laplace Transform of it I get after a theorem, which I have proved in my bachelor thesis, that

$$L(D^{4v}y(t))=s^{4/3}Y(s)-sD^{-2/3}y(0)-D^{1/3}y(0)=0 .$$

So $$Y(s)=\frac{sD^{-4/3}y(0)+D^{1/3}y(0)}{s^{4/3}}$$ And now? Actually I would have to do the inverse Laplace transform now, wouldn't I? But how do I do that? Apart from the fact that I haven't really understood the inverse Laplace transformation yet, I don't even know how to read it in a table or something. When I google I only come across the inverse Laplace transformation for ordinary differential equations and not for fractional differential equations.