Answer

Rearranging (ignore handling/conversion of constant),

\[ \begin{aligned} \difftwo{y} &= -\roundbr{1 + \frac{1}{y}}\roundbr{\diffone{y}}^{2}\newline \implies \frac{\difftwo{y}}{\diffone{y}} &= -\diffone{y} - \frac{\diffone{y}}{y}\newline \ln \diffone{y} &= -y - \ln y + c_{1}\newline \implies \diffone{y} &= \frac{c_{1}e^{-y}}{y}\newline e^{y}y\diffone{y} &= c_{1}\newline \implies ye^{y} - e^{y} &= c_{1}x + c_{2}\newline \end{aligned} \]

where the last equation is obtained after using integration by parts.