Question

how do you separate these variables dy/dx=4sqrt(y)lnx/x, y(e)=9

Answer 1

use the fact that
integral dy/dx dx = integral dy = y
find the antiderivative and plug in e and 9 to solve for the constant... i think...

Answer 2

dy/y^.5 = 4 ln x /x

2 y^.5 = 2 (ln x)^2

y^.5 = (ln x)^2 + C

y = (ln x)^4 + c

(e, 9) is on it
9 = 1^4 + c
c = 8
y = (ln x)^4 + 8