1. **State the problem:** Solve the differential equation $$ (2x - 5y + 1) \, dx + (-4y + 3x - 2) \, dy = 0 $$. 2. **Check if the equation is exact:** Let $$M = 2x - 5y + 1$$ and $$N = -4y + 3x - 2$$. Calculate partial derivatives: $$\frac{\partial M}{\partial y} = -5$$ $$\frac{\partial N}{\partial x} = 3$$ Since $$\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x}$$, the equation is not exact. 3. **Find an integrating factor:** Try an integrating factor depending on $$x$$ or $$y$$. Check $$\frac{\partial}{\partial y} \left(\frac{M}{N}\right)$$ or $$\frac{\partial}{\partial x} \left(\frac{N}{M}\right)$$ to find a suitable integrating factor. 4. **Try integrating factor $$\mu = e^{\int P(x) dx}$$ or $$\mu = e^{\int Q(y) dy}$$:** Calculate $$\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x} = -5 - 3 = -8$$. Calculate $$\frac{M}{N} = \frac{2x - 5y + 1}{-4y + 3x - 2}$$ (complicated, so try $$\mu(y)$$). 5. **Try integrating factor depending on $$x$$:** Calculate $$\frac{\partial}{\partial y} \left(\frac{M}{N}\right)$$ and check if it depends only on $$x$$. 6. **Alternatively, try integrating factor $$\mu = e^{\int \frac{\partial M/\partial y - \partial N/\partial x}{N} dy}$$:** Calculate $$\frac{-8}{-4y + 3x - 2}$$ which depends on both $$x$$ and $$y$$, so no simple integrating factor. 7. **Rewrite the equation:** $$ (2x - 5y + 1) \, dx + (-4y + 3x - 2) \, dy = 0 $$ Divide both sides by $$dx$$: $$ (2x - 5y + 1) + (-4y + 3x - 2) \frac{dy}{dx} = 0 $$ 8. **Solve for $$\frac{dy}{dx}$$:** $$ \frac{dy}{dx} = \frac{-(2x - 5y + 1)}{-4y + 3x - 2} = \frac{2x - 5y + 1}{4y - 3x + 2} $$ 9. **Try substitution:** Let $$v = \frac{y}{x}$$, so $$y = vx$$ and $$\frac{dy}{dx} = v + x \frac{dv}{dx}$$. 10. **Substitute into the equation:** $$ v + x \frac{dv}{dx} = \frac{2x - 5(vx) + 1}{4(vx) - 3x + 2} = \frac{2x - 5vx + 1}{4vx - 3x + 2} $$ Simplify numerator and denominator: $$ \frac{x(2 - 5v) + 1}{x(4v - 3) + 2} $$ 11. **Divide numerator and denominator by $$x$$:** $$ \frac{2 - 5v + \frac{1}{x}}{4v - 3 + \frac{2}{x}} $$ 12. **As $$x \to \infty$$, approximate:** $$ \frac{dy}{dx} \approx \frac{2 - 5v}{4v - 3} $$ 13. **Rewrite differential equation for $$v$$:** $$ v + x \frac{dv}{dx} = \frac{2 - 5v}{4v - 3} $$ 14. **Isolate $$\frac{dv}{dx}$$:** $$ x \frac{dv}{dx} = \frac{2 - 5v}{4v - 3} - v = \frac{2 - 5v - v(4v - 3)}{4v - 3} = \frac{2 - 5v - 4v^2 + 3v}{4v - 3} = \frac{2 - 2v - 4v^2}{4v - 3} $$ 15. **Simplify numerator:** $$ 2 - 2v - 4v^2 = -2(2v^2 + v - 1) $$ Factor quadratic: $$ 2v^2 + v - 1 = (2v - 1)(v + 1) $$ 16. **Rewrite:** $$ x \frac{dv}{dx} = \frac{-2(2v - 1)(v + 1)}{4v - 3} $$ 17. **Separate variables:** $$ \frac{4v - 3}{(2v - 1)(v + 1)} dv = -2 \frac{dx}{x} $$ 18. **Integrate both sides:** $$ \int \frac{4v - 3}{(2v - 1)(v + 1)} dv = -2 \int \frac{dx}{x} $$ 19. **Use partial fractions for left integral:** Set $$ \frac{4v - 3}{(2v - 1)(v + 1)} = \frac{A}{2v - 1} + \frac{B}{v + 1} $$ Multiply both sides by denominator: $$ 4v - 3 = A(v + 1) + B(2v - 1) $$ 20. **Solve for A and B:** For $$v = \frac{1}{2}$$: $$ 4(\frac{1}{2}) - 3 = A(\frac{1}{2} + 1) + B(0) \Rightarrow 2 - 3 = A(\frac{3}{2}) \Rightarrow -1 = \frac{3}{2} A \Rightarrow A = -\frac{2}{3} $$ For $$v = -1$$: $$ 4(-1) - 3 = A(0) + B(2(-1) - 1) \Rightarrow -4 - 3 = B(-2 - 1) \Rightarrow -7 = -3B \Rightarrow B = \frac{7}{3} $$ 21. **Rewrite integral:** $$ \int \left( \frac{-\frac{2}{3}}{2v - 1} + \frac{\frac{7}{3}}{v + 1} \right) dv = -2 \ln|x| + C $$ 22. **Integrate:** $$ -\frac{2}{3} \int \frac{dv}{2v - 1} + \frac{7}{3} \int \frac{dv}{v + 1} = -2 \ln|x| + C $$ 23. **Integrate each term:** $$ -\frac{2}{3} \cdot \frac{1}{2} \ln|2v - 1| + \frac{7}{3} \ln|v + 1| = -2 \ln|x| + C $$ Simplify: $$ -\frac{1}{3} \ln|2v - 1| + \frac{7}{3} \ln|v + 1| = -2 \ln|x| + C $$ 24. **Multiply both sides by 3:** $$ -\ln|2v - 1| + 7 \ln|v + 1| = -6 \ln|x| + C' $$ 25. **Rewrite as:** $$ \ln \left| \frac{(v + 1)^7}{2v - 1} \right| = \ln \left| \frac{C''}{x^6} \right| $$ 26. **Exponentiate both sides:** $$ \frac{(v + 1)^7}{2v - 1} = \frac{C}{x^6} $$ 27. **Recall substitution $$v = \frac{y}{x}$$:** $$ \frac{\left( \frac{y}{x} + 1 \right)^7}{2 \frac{y}{x} - 1} = \frac{C}{x^6} $$ Multiply numerator and denominator by $$x$$: $$ \frac{(y + x)^7}{x^7 (2y - x)} = \frac{C}{x^6} $$ 28. **Multiply both sides by $$x^7 (2y - x)$$:** $$ (y + x)^7 = C x (2y - x) $$ **Final implicit solution:** $$ (y + x)^7 = C x (2y - x) $$