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Let y = y(x) solve sin x (dy/dx) + y cos x = 4x on (0, pi). If y(pi/2) = 0, find y(pi/6).

1. -8/(9 sqrt(3)) * pi²
2. -8/9 * pi²
3. -4/9 * pi²
4. 4/(9 sqrt(3)) * pi²

Correct answer: -8/9 * pi²

Solution

Since sin x y' + cos x y = (y sin x)', integration gives y sin x = 2x² + C; the condition fixes C and dividing by sin(pi/6)=1/2 yields y(pi/6) = -8 pi²/9.

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