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JEE Main 2025 · Mathematics · Differential Equations · hard

If y(x) satisfies dy/dx+ytan x=sin 2x with y(0)=1, then y ((π)/(3)) equals

Single correct option

If y(x)y(x) satisfies dydx+ytan⁡x=sin⁡2x\dfrac{dy}{dx}+y\tan x=\sin 2x with y(0)=1y(0)=1, then y ⁣(π3)y\!\left(\dfrac{\pi}{3}\right) equals
  1. A. 12\tfrac12
  2. B. 32\tfrac32
  3. C. 14\tfrac{1}{4}
  4. D. 11
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Answer: (D) 11

The integrating factor is e∫tan⁡x dx=sec⁡xe^{\int\tan x\,dx}=\sec x. Then ysec⁡x=∫sin⁡2xsec⁡x dx=∫2sin⁡x dx=−2cos⁡x+Cy\sec x=\int\sin2x\sec x\,dx=\int2\sin x\,dx=-2\cos x+C, so y=−2cos⁡2x+Ccos⁡xy=-2\cos^2x+C\cos x. From y(0)=1y(0)=1: −2+C=1-2+C=1, C=3C=3. At x=π/3x=\pi/3: y=−2⋅14+3⋅12=1y=-2\cdot\tfrac14+3\cdot\tfrac12=1.
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