1. The problem is to solve the differential equation $$\frac{dy}{dx} - y = e^x y^2$$. 2. This is a nonlinear first-order differential equation. We can try to solve it by substitution or by recognizing it as a Bernoulli equation. 3. The Bernoulli equation has the form $$\frac{dy}{dx} + P(x)y = Q(x)y^n$$. Here, rewrite the equation as $$\frac{dy}{dx} - y = e^x y^2$$ or $$\frac{dy}{dx} + (-1) y = e^x y^2$$, so $$P(x) = -1$$, $$Q(x) = e^x$$, and $$n=2$$. 4. To solve Bernoulli equations, use the substitution $$v = y^{1-n} = y^{1-2} = y^{-1}$$. 5. Differentiate $$v$$ with respect to $$x$$: $$\frac{dv}{dx} = -y^{-2} \frac{dy}{dx}$$. 6. Rearranged, $$\frac{dy}{dx} = -y^{2} \frac{dv}{dx}$$. 7. Substitute into the original equation: $$-y^{2} \frac{dv}{dx} - y = e^x y^{2}$$. 8. Divide both sides by $$y^{2}$$: $$-\frac{dv}{dx} - \frac{y}{y^{2}} = e^x$$ or $$-\frac{dv}{dx} - y^{-1} = e^x$$. 9. Recall $$v = y^{-1}$$, so: $$-\frac{dv}{dx} - v = e^x$$. 10. Multiply both sides by $$-1$$: $$\frac{dv}{dx} + v = -e^x$$. 11. This is a linear first-order ODE in $$v$$. The integrating factor is: $$\mu(x) = e^{\int 1 dx} = e^x$$. 12. Multiply both sides by $$e^x$$: $$e^x \frac{dv}{dx} + e^x v = -e^{2x}$$. 13. Left side is derivative of $$v e^x$$: $$\frac{d}{dx} (v e^x) = -e^{2x}$$. 14. Integrate both sides: $$v e^x = \int -e^{2x} dx + C = -\frac{1}{2} e^{2x} + C$$. 15. Solve for $$v$$: $$v = e^{-x} \left(C - \frac{1}{2} e^{2x} \right) = C e^{-x} - \frac{1}{2} e^{x}$$. 16. Recall $$v = y^{-1}$$, so: $$y^{-1} = C e^{-x} - \frac{1}{2} e^{x}$$. 17. Finally, solve for $$y$$: $$y = \frac{1}{C e^{-x} - \frac{1}{2} e^{x}} = \frac{1}{C e^{-x} - \frac{1}{2} e^{x}}$$. This is the general solution to the differential equation.