solve the initial value problem by separation of variables 8. dy/dx=x+1/xy, x>0, y(1)=-4

Answer 1

Please check that you have sufficient parentheses.
The problem posted (implicitly) is
dy/dx = x+ (1/xy) which is not easily separable.

I assume you wished to post:
dy/dx = (x+1)/xy
which can be separated as:
ydy = (x+1)dx/x = (1+1/x)dx
which can be easily integrated on both sides. The constant of integration can be obtained by substitution of the initial condition y(1)=-4.