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Let y(x) satisfy the differential equation y' - y tan x = 2x sec x with y(0) = 0. Which of the following statements is correct?

1. y(pi/4) = pi²/(8 sqrt(2))
2. y'(pi/4) = pi²/18
3. y(pi/4) = pi²/9
4. y'(pi/3) = 4pi/3 + 2pi²/(3 sqrt(3))

Correct answer: y(pi/4) = pi²/(8 sqrt(2))

Solution

The integrating factor cos x turns the equation into (y cos x)' = 2x, giving y = x²/cos x; evaluating at pi/4 gives pi²/(8 sqrt 2).

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