Just minutes ago I asked the wrong question. My apologies to the moderators.
If it was the second derivative instead of third then it would be the Airy equation.
The integral representation is simple to see through Fourier transform but I wonder if there is a name and solution for this type of equations. Actually, I am also interested in its general form:
$y^{(n)}=xy$
Here $n$ denotes $n$'th derivative.