Solve : $ y dx +x dy +\sin y \cos ^2 (xy) dy+e^x \cos^2 (xy) dx=0$

Question

$(a)\;\tan xy -\cos y=c \\ (b)\;\tan xy+e^x-\cos y=c \\ (c)\;e^x-\tan xy=c \\ (d)\;\sec xy-e^x- \sin y =c $

Answer 1

divide by $\cos^2 (xy)$

$\large\frac{y dx+x dy}{\cos^2 (xy)}$$+\sin y dy +e^x dx=0$

$\large\frac{d(xy)}{\cos^2 (xy)}$$+\sin y dy +e^x dx=0$

$\sec^2 (xy) d(xy) +e^x dx+ \sin y dy=0$

Integrate:

$\tan xy +e^x - \cos y =c$

Hence b is the correct answer.