Uhm, let me see if I know what you're asking.

$\displaystyle y'' = -ay + \frac{b}{y^2} $

$\displaystyle y' = -a \int {y} dy + b \int {y^{-2}} dy $

$\displaystyle y' = -a(\frac {y^2}{2} + C_1) + b(-y^{-1} + C_2) $

$\displaystyle y' = -a\frac {y^2}{2} - by^{-1} + C_t $

$\displaystyle y = -a \int (\frac {y^2}{2})dy - b \int (y^{-1})dy + \int C_t dy$

$\displaystyle y = -a (\frac {y^3}{6} + C_3) - b (\ln y + C_4) + C_ty + C_5 $

$\displaystyle y = - \frac {ay^3}{6} - b \ln y + C_ty + C_T $

It's either that or I'm totally off the mark.