$b^2 + y^2 - 2y + z + y = c$ and $xdy - 2ydy - dy = 2xdx - ydx + dx$
Step-by-Step Solution
Key Concept: Recognize exact differentials like $d(xy) = xdy + ydx$ to simplify and integrate the differential equation.
Starting from the differential equation $(xdy + ydx) - 2ydy - dy = 2xdx - ydx + dx$, we rearrange to get $d(xy) - 2ydy - dy = 2xdx - ydx + dx$. Simplifying: $(xdy + ydx) - dy(2y + 1) = dx(2x - y + 1)$. After integration, we obtain $xy - y^2 - y - x^2 - x = c$.
Correct Answer: 2