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Let y = y(x) satisfy the differential equation dy/dx = |A|, for all x > 0, where A is the 3x3 matrix [[y, sin(x), 1], [0, -1, 1], [2, 0, 1/x]]. Given that y(pi) = pi + 2, find the value of y(pi/2).
- pi/2 + 4/pi
- pi/2 - 1/pi
- 3*pi/2 - 1/pi
- pi/2 - 4/pi
Correct answer: pi/2 + 4/pi
Solution
Expanding det(A) gives -y/x + 2*sin(x) + 2. The ODE dy/dx = -y/x + 2*sin(x) + 2 rearranges to dy/dx + y/x = 2*sin(x) + 2, a linear first-order equation. With integrating factor x, the solution is xy = -2x*cos(x) + 2*sin(x) + x² + C. Using y(pi) = pi+2 gives C = 0, and evaluating at x = pi/2 yields y(pi/2) = pi/2 + 4/pi.
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