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Let y = y(x) be the solution of the differential equation x*dy = (y + x³ * cos(x)) dx, with y(pi) = 0. Find y(pi/2).
- pi²/4 + pi/2
- pi²/2 + pi/4
- pi²/2 - pi/4
- pi²/4 - pi/2
Correct answer: pi²/4 - pi/2
Solution
The ODE x dy = (y + x³ cos x) dx can be written dy/dx - y/x = x² cos x (linear, first order). IF = e^(integral -1/x dx) = e^(-ln x) = 1/x. Multiply: d(y/x)/dx = x cos x. Integrate: y/x = integral(x cos x dx) = x sin x + cos x + C. Apply y(pi)=0: 0 = pi*sin(pi) + cos(pi) + C = 0 - 1 + C => C = 1. So y = x(x sin x + cos x + 1). At x=pi/2: y = (pi/2)((pi/2)*sin(pi/2) + cos(pi/2) + 1) = (pi/2)(pi/2*1 + 0 + 1) = (pi/2)(pi/2 + 1) = pi²/4 + pi/2.
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