1. **Problem:** Solve the differential equation $$\frac{dy}{dx} - \frac{y}{x} = xy^2$$ using Bernoulli's equation. 2. **Bernoulli's equation form:** $$\frac{dy}{dx} + P(x)y = Q(x)y^n$$ where $n \neq 0,1$. 3. **Identify terms:** Here, $$P(x) = -\frac{1}{x}, \quad Q(x) = x, \quad n = 2.$$ 4. **Substitution:** Let $$v = y^{1-n} = y^{1-2} = y^{-1}.$$ Then, $$\frac{dv}{dx} = -y^{-2} \frac{dy}{dx}.$$ 5. **Rewrite original equation:** Multiply both sides by $-y^{-2}$: $$-y^{-2} \frac{dy}{dx} + \frac{1}{x} y^{-1} = -x.$$ This becomes: $$\frac{dv}{dx} + \frac{1}{x} v = -x.$$ 6. **This is a linear ODE in $v$: $$\frac{dv}{dx} + \frac{1}{x} v = -x.$$** 7. **Integrating factor:** $$\mu(x) = e^{\int \frac{1}{x} dx} = e^{\ln|x|} = x.$$ 8. **Multiply entire equation by $\mu(x)$:** $$x \frac{dv}{dx} + v = -x^2.$$ 9. **Left side is derivative:** $$\frac{d}{dx} (x v) = -x^2.$$ 10. **Integrate both sides:** $$x v = -\frac{x^3}{3} + C.$$ 11. **Solve for $v$:** $$v = -\frac{x^2}{3} + \frac{C}{x}.$$ 12. **Recall substitution:** $$v = y^{-1} = \frac{1}{y}.$$ 13. **Final solution:** $$\frac{1}{y} = -\frac{x^2}{3} + \frac{C}{x} \implies y = \frac{1}{-\frac{x^2}{3} + \frac{C}{x}} = \frac{1}{\frac{C}{x} - \frac{x^2}{3}}.$$ This is the general solution to the first differential equation using Bernoulli's method.