1. **State the problem:** Solve the differential equation $$\left(1-\frac{3}{y}+x\right)\frac{dy}{dx}+y=\frac{3}{x}-1$$ for $y$ as a function of $x$. 2. **Rewrite the equation:** We want to isolate $\frac{dy}{dx}$: $$\left(1-\frac{3}{y}+x\right)\frac{dy}{dx} = \frac{3}{x} - 1 - y$$ 3. **Express $\frac{dy}{dx}$ explicitly:** $$\frac{dy}{dx} = \frac{\frac{3}{x} - 1 - y}{1 - \frac{3}{y} + x}$$ 4. **Simplify the denominator:** Rewrite $1 - \frac{3}{y} + x$ as $\frac{y - 3 + xy}{y} = \frac{y(x+1) - 3}{y}$. 5. **Rewrite $\frac{dy}{dx}$ using this:** $$\frac{dy}{dx} = \frac{\frac{3}{x} - 1 - y}{\frac{y(x+1) - 3}{y}} = \left(\frac{3}{x} - 1 - y\right) \cdot \frac{y}{y(x+1) - 3}$$ 6. **Rewrite numerator:** $$\frac{3}{x} - 1 - y = \frac{3 - x - xy}{x}$$ 7. **Substitute back:** $$\frac{dy}{dx} = \frac{3 - x - xy}{x} \cdot \frac{y}{y(x+1) - 3} = \frac{y(3 - x - xy)}{x(y(x+1) - 3)}$$ 8. **Rewrite numerator:** $$3 - x - xy = 3 - x(1 + y)$$ 9. **Rewrite denominator:** $$y(x+1) - 3 = y(x+1) - 3$$ (already simplified) 10. **Final form:** $$\frac{dy}{dx} = \frac{y(3 - x(1 + y))}{x(y(x+1) - 3)}$$ 11. **Separate variables if possible:** Rewrite as $$\frac{y(x+1) - 3}{y(3 - x(1 + y))} dy = \frac{1}{x} dx$$ 12. **This is a separable form.** Integrate both sides: $$\int \frac{y(x+1) - 3}{y(3 - x(1 + y))} dy = \int \frac{1}{x} dx$$ 13. **However, the integral is complicated due to mixed variables.** This suggests the equation is not separable in the usual way. 14. **Alternative approach:** Rewrite original equation as $$\left(1 - \frac{3}{y} + x\right) dy + \left(y - \frac{3}{x} + 1\right) dx = 0$$ 15. **Check if exact:** Let $$M = 1 - \frac{3}{y} + x, \quad N = y - \frac{3}{x} + 1$$ Compute partial derivatives: $$\frac{\partial M}{\partial x} = 1, \quad \frac{\partial N}{\partial y} = 1$$ Since $\frac{\partial M}{\partial x} = \frac{\partial N}{\partial y}$, the equation is exact. 16. **Find potential function $\Psi(x,y)$ such that:** $$\frac{\partial \Psi}{\partial y} = M = 1 - \frac{3}{y} + x$$ Integrate w.r.t. $y$: $$\Psi = y - 3 \ln|y| + xy + h(x)$$ 17. **Differentiate $\Psi$ w.r.t. $x$ and set equal to $N$:** $$\frac{\partial \Psi}{\partial x} = y + h'(x) = y - \frac{3}{x} + 1$$ Solve for $h'(x)$: $$h'(x) = - \frac{3}{x} + 1$$ 18. **Integrate $h'(x)$:** $$h(x) = -3 \ln|x| + x + C$$ 19. **Write implicit solution:** $$\Psi(x,y) = y - 3 \ln|y| + xy - 3 \ln|x| + x = C$$ 20. **Final answer:** $$\boxed{y - 3 \ln|y| + xy - 3 \ln|x| + x = C}$$ This implicit solution satisfies the original differential equation.