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Solve the differential equation (x² sin³ y - y² cos x) dx + (x³ cos y sin² y - 2y sin x) dy = 0.

1. x³ sin³ y = 3y² sin x + C
2. x³ sin³ y + 3y² sin x = C
3. x² sin³ y + y³ sin x = C
4. 2x² sin y + y² sin x = C

Correct answer: x³ sin³ y = 3y² sin x + C

Solution

The equation is exact; integrating gives x³ sin³ y/3 - y² sin x = const, i.e. x³ sin³ y - 3y² sin x = C.

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