1. **Problem Statement:** Solve the differential equation $ (2x - y) dx - (x + 4y) dy = 0 $ and find the implicit solution. 2. **Identify the type:** This is a first order differential equation in differential form $M dx + N dy = 0$ where $M = 2x - y$ and $N = -(x + 4y)$. 3. **Check if exact:** Compute $\frac{\partial M}{\partial y} = -1$ and $\frac{\partial N}{\partial x} = -1$. Since $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$, the equation is exact. 4. **Find potential function $\psi(x,y)$:** - Integrate $M$ with respect to $x$: $$\psi(x,y) = \int (2x - y) dx = x^2 - xy + h(y)$$ - Differentiate $\psi$ with respect to $y$: $$\frac{\partial \psi}{\partial y} = -x + h'(y)$$ - Set equal to $N$: $$-x + h'(y) = -(x + 4y) \implies h'(y) = -4y$$ - Integrate $h'(y)$: $$h(y) = -2y^2 + C$$ 5. **Write implicit solution:** $$\psi(x,y) = x^2 - xy - 2y^2 = C$$ 6. **Match with options:** The implicit solution matches option $2x^2 - 2xy - y^2 = c$ after multiplying entire equation by 2: $$2(x^2 - xy - 2y^2) = 2x^2 - 2xy - 4y^2 = c'$$ But none exactly match $x^2 - xy - 2y^2 = c$. Check options carefully: - Option 2: $2x^2 - 2xy - y^2 = c$ differs in $y^2$ coefficient. - Option 3: $y^2 - xy + 2x^2 = c$ rearranged is $2x^2 - xy + y^2 = c$ which is close but signs differ. Re-examining the integration: The found solution is $x^2 - xy - 2y^2 = c$. Multiply both sides by $-1$: $$-x^2 + xy + 2y^2 = -c$$ Rearranged: $$2y^2 + xy - x^2 = c$$ This matches option 1: $2y^2 + xy - x^2 = c$. **Final answer:** $2y^2 + xy - x^2 = c$.