Q:

why $xy^{(4)} + 2y'' + (xy')^5 = x^3$ is of the fourth order differential equation

I came across this equation from the book Ordinary Differential Equations (Lesson 3A page 31) 
$$xy^{(4)} + 2y'' + (xy')^5 = x^3$$ 
I don't know why it was regarded as a fourth order differential equation. Or does it mean that $$xy^{(4)} = xy''''$$ 
I am a bit confused

A:

You are correct that $y^{(4)}=y''''$. For higher derivatives the latter notation becomes messy and hard to read, which is why it is common to see $y^{(n)}$ if $n$ becomes large, but also $y'$ and $y''$ in the same equation.
See also: Lagrange's notation.

