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Let y = y(x) be the solution of the differential equation sin x dy/dx + y cos x = 4x, x ∈ (0, π). If y(π/2) = 0, then y(π/6) is equal to:
- −8π²/(9√3)
- −8π²/9
- −4π²/9
- 4π²/(9√3)
Correct answer: −8π²/9
Solution
The equation is d/dx(y sin x) = 4x, so y sin x = 2x^2 + C. Using y(pi/2)=0 gives C = -pi^2/2. At x=pi/6: y*(1/2) = 2(pi/6)^2 - pi^2/2 = -4pi^2/9, so y(pi/6) = -8pi^2/9.
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