1. **State the problem:** Solve the differential equation $$(6x - 3y + 2)dx + (2x - y - 1)dy = 0.$$ 2. **Identify the type of equation:** This is a first-order differential equation in the form $$M(x,y)dx + N(x,y)dy = 0,$$ where $$M = 6x - 3y + 2$$ and $$N = 2x - y - 1.$$ 3. **Check if the equation is exact:** Calculate $$\frac{\partial M}{\partial y}$$ and $$\frac{\partial N}{\partial x}.$$ - $$\frac{\partial M}{\partial y} = -3$$ - $$\frac{\partial N}{\partial x} = 2$$ Since $$\frac{\partial M}{\partial y} \neq \frac{\partial N}{\partial x},$$ the equation is not exact. 4. **Find an integrating factor:** Try an integrating factor depending on $$x$$ or $$y$$. - Compute $$\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} = 2 - (-3) = 5.$$ - Compute $$\frac{M}{N} = \frac{6x - 3y + 2}{2x - y - 1}$$ which is not a function of $$x$$ or $$y$$ alone, so try $$\mu(y)$$ or $$\mu(x)$$. - Check if $$\frac{\frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}{M} = \frac{5}{6x - 3y + 2}$$ is a function of $$x$$ alone (no), or if $$\frac{\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}}{N} = \frac{-5}{2x - y - 1}$$ is a function of $$y$$ alone (no). 5. **Try an integrating factor depending on $$x$$ and $$y$$:** Since simple methods fail, try to solve by rearranging or substitution. 6. **Rewrite the equation:** $$ (6x - 3y + 2)dx + (2x - y - 1)dy = 0 $$ Divide both sides by $$dx$$ (assuming $$dx \neq 0$$): $$ (6x - 3y + 2) + (2x - y - 1) \frac{dy}{dx} = 0 $$ So, $$ \frac{dy}{dx} = - \frac{6x - 3y + 2}{2x - y - 1} $$ 7. **Try substitution:** Let $$v = y/x$$, so $$y = vx$$ and $$\frac{dy}{dx} = v + x \frac{dv}{dx}.$$ Substitute into the equation: $$ v + x \frac{dv}{dx} = - \frac{6x - 3(vx) + 2}{2x - v x - 1} = - \frac{6x - 3vx + 2}{2x - vx - 1} $$ Simplify numerator and denominator: Numerator: $$6x - 3vx + 2 = x(6 - 3v) + 2$$ Denominator: $$2x - vx - 1 = x(2 - v) - 1$$ So, $$ v + x \frac{dv}{dx} = - \frac{x(6 - 3v) + 2}{x(2 - v) - 1} $$ 8. **Isolate $$x \frac{dv}{dx}$$:** $$ x \frac{dv}{dx} = - \frac{x(6 - 3v) + 2}{x(2 - v) - 1} - v $$ 9. **This is a complicated expression; however, the substitution reduces the problem to a separable or solvable form.** Further algebraic manipulation or numerical methods may be needed to solve explicitly. **Final answer:** The differential equation can be transformed using substitution $$v = \frac{y}{x}$$ into $$ x \frac{dv}{dx} = - \frac{x(6 - 3v) + 2}{x(2 - v) - 1} - v, $$ which can be further analyzed or solved by advanced methods.