Given $\displaystyle dy/dx = -\frac{8y^2}{(x-1)^3}$ at point (x,y)
and the curve goes through (2, -1/3)
Prove the equation of the curve,
$\displaystyle y=\frac{x^2-2x+1}{x^2-2x-3}$
Separate the variables and integrate...
$\displaystyle \displaystyle \int \dfrac{1}{y^2}\ dy =-8 \int \dfrac{1}{(x-1)^3}\ dx$
Then, use the values of (2, -1/3) to find the value of the constant of integration.
Or, you can find directly:
$\displaystyle \displaystyle \int^y_{-\frac13} \dfrac{1}{y^2}\ dy =-8 \int^x_2 \dfrac{1}{(x-1)^3}\ dx$
EDIT: You should know by now that you should post what you have tried.
Have you tried anything your self? To show that a given function satisifies a differential equation is typically much simpler than actually solving the equation- just differentiate the given y and show that it satisfies the equation.
However, in this case the equation "separates" as $\displaystyle y^{-2}dy= -\frac{8 dx}{(x- 1)^3}$ and those are easy to integrate.