I'd like to strongly suggest the Phys.SE community to carefully re-examine the following question and the reasons it was put on hold or regarded as not a good enough question:
OP's question about the potential energy being $|x|^3$ but it then described as simple harmonic motion seem completely legitimate to me.
The solution from the book doesn't satisfy the equation it is supposed to solve, as OP said there very clearly. What effort could OP be expected to make, for example explicitly trying and failing to solve the differential equation using the wrong solution? I don't think there is a reasonable effort lacking here, on the part of OP. As to the people voting to close the question and calling it homework, at the risk of being rude, I'm far from impressed about their level of effort.
It also clearly isn't a homework-like problem. OP asked whether that is a case of simple harmonic motion as the textbook says. The answer is that it isn't, and that the textbook he showed the image from is wrong.