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Let y = y(x) be the solution of the differential equation cos(x) * (dy/dx) + y*sin(x) = 1 for x in (0, pi/2), with the initial condition y(pi/4) = 0. Find the value of y(pi/6).
- (3 + sqrt(3))/2
- (3 - sqrt(3))/4
- (1 - sqrt(3))/2
- 1 - sqrt(3)
Correct answer: (1 - sqrt(3))/2
Solution
Multiplying both sides by sec(x) converts the equation to d/dx(y*sec x) = sec²(x). Integrating gives y*sec(x) = tan(x) + C. Applying y(pi/4) = 0 yields C = -1, so y = sin(x) - cos(x). At x = pi/6: y = 1/2 - sqrt(3)/2 = (1 - sqrt(3))/2.
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