Let g be a differentiable function such that ∫0xg(t)dt=x−∫0xtg(t)dt,x≥0 and let y=y(x) satisfy the differential equation dxdy−ytanx= 2(x+1)secxg(x),x∈[0,2π). If y(0)=0, then y(3π) is equal to
Held on 3 Apr 2025 · Verified 6 Jul 2026.
332π
34π
32π
334π
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Let $y = y(x)$ be the solution of the differential equation $x\sin\left(\dfrac{y}{x}\right)dy = \left(y\sin\left(\dfrac{y}{x}\right) - x\right)dx$, $y(1) = \dfrac{\pi}{2}$ and let $\alpha = \cos\left(\dfrac{y(e^{12})}{e^{12}}\right)$. Then the number of integral values of $p$, for which the equation $x^2 + y^2 - 2px + 2py + \alpha + 2 = 0$ represents a circle of radius $r \leq 6$, is __________.
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