1. **State the problem:** Solve the differential equation $$\frac{dy}{dx} - y = e^x y^2$$ by substitution. 2. **Rewrite the equation:** The equation is nonlinear due to the $$y^2$$ term. We can try the substitution $$v = \frac{1}{y}$$ to simplify it. 3. **Find $$\frac{dv}{dx}$$:** Since $$v = \frac{1}{y}$$, then $$y = \frac{1}{v}$$ and by differentiating both sides with respect to $$x$$: $$\frac{dy}{dx} = -\frac{1}{v^2} \frac{dv}{dx}$$ 4. **Substitute into the original equation:** Replace $$\frac{dy}{dx}$$ and $$y$$: $$-\frac{1}{v^2} \frac{dv}{dx} - \frac{1}{v} = e^x \left(\frac{1}{v}\right)^2 = \frac{e^x}{v^2}$$ 5. **Multiply through by $$v^2$$ to clear denominators:** $$-\frac{dv}{dx} - v = e^x$$ 6. **Rewrite the equation:** $$\frac{dv}{dx} + v = -e^x$$ 7. **Solve the linear first-order ODE:** The integrating factor is $$\mu(x) = e^{\int 1 dx} = e^x$$. Multiply both sides by $$e^x$$: $$e^x \frac{dv}{dx} + e^x v = -e^{2x}$$ 8. **Recognize the left side as a derivative:** $$\frac{d}{dx} (v e^x) = -e^{2x}$$ 9. **Integrate both sides:** $$v e^x = \int -e^{2x} dx = -\frac{1}{2} e^{2x} + C$$ 10. **Solve for $$v$$:** $$v = -\frac{1}{2} e^x + C e^{-x}$$ 11. **Recall substitution $$v = \frac{1}{y}$$:** $$\frac{1}{y} = -\frac{1}{2} e^x + C e^{-x}$$ 12. **Final solution:** $$y = \frac{1}{-\frac{1}{2} e^x + C e^{-x}} = \frac{1}{C e^{-x} - \frac{1}{2} e^x}$$ This is the general solution to the differential equation by substitution.