Is this a typo or alternative notation? I am reading Calculus:Volume II by Chen, Liu, etc. 
on page 201, the book states this. 

...For example, $$ y^{\prime\prime} +xy^{\prime 3} + 5y = e^x $$ is a third-order equation. 

I have seen $y^{(3)}$ as meaning the third derivative of y in prime notation; however, I have never seen $y^{\prime N}$ where $N$ is a natural numer for the $N$th derivative of a variable. 

This is definitely a second order equation and $y'^3$ means the cube of $y'$.
