1. The problem is to solve the equation $$ye^{-3x} - 3x = y^2$$ for $y$ in terms of $x$. 2. This is a nonlinear equation involving both $y$ and $x$. We want to isolate $y$ if possible. 3. Rewrite the equation: $$ye^{-3x} - y^2 = 3x$$ 4. Factor $y$ terms: $$y e^{-3x} - y^2 = y(e^{-3x} - y) = 3x$$ 5. This is a quadratic in $y$ if rearranged: $$y^2 - y e^{-3x} + 3x = 0$$ 6. Use the quadratic formula for $y$: $$y = \frac{e^{-3x} \pm \sqrt{(e^{-3x})^2 - 4 \cdot 1 \cdot 3x}}{2} = \frac{e^{-3x} \pm \sqrt{e^{-6x} - 12x}}{2}$$ 7. The solution for $y$ depends on the discriminant $$e^{-6x} - 12x$$ being non-negative for real $y$. 8. Therefore, the solutions are: $$y = \frac{e^{-3x} \pm \sqrt{e^{-6x} - 12x}}{2}$$ This completes solving for $y$ in terms of $x$.