1. **State the problem:** Solve the differential equation $ (x+y) \, dx + x \, dy = 0 $. 2. **Rewrite the equation:** Express it in the form $ M(x,y) \, dx + N(x,y) \, dy = 0 $ where $ M = x + y $ and $ N = x $. 3. **Check if the equation is exact:** Calculate $ \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 the potential function $ \psi(x,y) $:** Since $ \frac{\partial \psi}{\partial x} = M = x + y $, integrate with respect to $ x $: $$ \psi(x,y) = \int (x + y) \, dx = \frac{x^2}{2} + xy + h(y) $$ 5. **Find $ h(y) $:** Differentiate $ \psi $ with respect to $ y $: $$ \frac{\partial \psi}{\partial y} = x + h'(y) $$ Set equal to $ N = x $: $$ x + h'(y) = x \implies h'(y) = 0 $$ So, $ h(y) $ is a constant. 6. **Write the implicit solution:** $$ \psi(x,y) = \frac{x^2}{2} + xy = C $$ where $ C $ is an arbitrary constant. **Final answer:** $$ \frac{x^2}{2} + xy = C $$