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JEE Main 2017

JEE Main 2017 Mathematics question on Differential Equations

If (2+sin⁡x)dydx+(y+1)cos⁡x=0(2 + \sin x)\frac{dy}{dx} + (y+1)\cos x = 0 and y(0)=1y(0) = 1, then y(π2)y\left(\frac{\pi}{2}\right) is equal to
  1. A.
    13\frac{1}{3}
  2. B.
    −23-\frac{2}{3}
  3. C.
    −13-\frac{1}{3}
  4. D.
    43\frac{4}{3}
Show answer and solution

Correct option: A

Solution

Separate variables: dy/(y+1) = -cos x dx/(2+sin x).

Integrate both sides: ln|y+1| = -ln|2+sin x| + C, so (y+1)(2+sin x) = K.

Apply y(0)=1: (1+1)(2+0) = K → K = 4.

At x = π/2: (y+1)(2+1) = 4 → y+1 = 4/3 → y = 1/3.

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