(A) −7/4

(B) −3/4

(C) 3/4

(D) 7/4

Correct Answer: 3/4

Explanation

Given differential equation:

x dy/dx − sin 2y = x³(2 − x³)cos²y

Use identity sin 2y = 2 sin y cos y. Divide the entire equation by cos²y:

x (sec²y) dy/dx − 2 tan y = x³(2 − x³)

Let t = tan y. Then dt/dx = sec²y · dy/dx.

x dt/dx − 2t = x³(2 − x³)

This is a linear differential equation:

dt/dx − (2/x)t = x²(2 − x³)

Integrating factor:

IF = e^{∫(−2/x)dx} = x^{−2}

Multiplying throughout:

d/dx (t/x²) = 2 − x³

Integrate:

t/x² = 2x − x⁴/4 + C

So,

t = 2x³ − x⁶/4 + Cx²

Using condition y(2) = 0 ⇒ tan y(2) = 0:

0 = 2(8) − 64/4 + 4C 0 = 16 − 16 + 4C C = 0

Thus,

tan y = 2x³ − x⁶/4

At x = 1:

tan(y(1)) = 2 − 1/4 = 3/4

Hence, the required value is 3/4.

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