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Find the general solution of the differential equation log x (dy)/(dx) + (y)/(x) = sin 2x.

1. ylog|x| = C - (1)/(2)cos x
2. ylog|x| = C + (1)/(2)cos 2x
3. ylog|x| = C - (1)/(2)cos 2x
4. xylog|x| = C - (1)/(2)cos 2x

Correct answer: ylog|x| = C - (1)/(2)cos 2x

Solution

(log x) dy/dx + y/x = d/dx[y log x] = sin 2x. Integrating: y log|x| = -1/2 cos 2x + C, i.e. y log|x| = C - 1/2 cos 2x.

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