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. 

Here is the terminology employed for algebraic differential equations
$$\rm  a\, y^{e_0} (y')^{e_1}\cdots (y^{(k)}\!)^{e_k} $$


*

*has $ $ order $\rm\,\ k$

*has $ $ degree $\rm\,\ e_0 + e_1 + \cdots + e_k$

*has $ $ weight $\rm\,\ e_0 + 2 e_2 + \cdots + (k+1) e_k$
A differential polynomial is a sum of such differential monomials. It's order is the maximum order of any of its monomials. Similarly for its degree and weight.
