Rather than using the formal condition dM/dy = $\frac{dN}{dx}$, JEE problems are best solved by recognizing standard exact differential patterns. The key patterns: (1) x dy + y dx = d(xy) — appears whenever you see terms like xdy + ydx grouped together. (2) x dy - y dx = $x^2$*d$\frac{y}{x}$ — divide both sides by $x^2$ to get d$\frac{y}{x}$. Equivalently = -$y^2$*d$\frac{x}{y}$. (3) $\frac{x dy - y dx}{(x^2 + y^2)}$ = d(arctan$\frac{y}{x}$) — appears in polar-related DEs. (4) $\frac{x dx + y dy}{(x^2 + y^2)}$ = d(ln(sqrt(x^{2+y}^2)))/1 — related to distance from origin. (5) $e^x$(f + f') dx = d($e^x$*f(x)) — very common pattern in JEE. Recognizing these patterns can turn a seemingly complex DE into a one-step solution.