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JEE Main202523 Jan 2025Morning ShiftMathematicsDifferential EquationsActual

Let a curve y=f(x) pass through the points (0,5) and ( _e 2, k ) . If the curve satisfies the differential equation 2(3+y) e^ 2 x d x- (7+e^ 2 x ) d y=0 , then k is equal to

Options

  1. A4
  2. B32
  3. C8
  4. D16

Correct answer

C. 8

Step-by-step solution

aligned & d y d x = 2(3+y) e^ 2 x 7+e^ 2 x & d y d x - 2 y e^ 2 x 7+e^ 2 x = 6 e^ 2 x 7+e^ 2 x & I.F. =e^ - 2 e^ 2 x 7+e^ 2 x d x & e^ - (7+e^ 2 x ) & = 1 7+e^ 2 x & y 1 7+e^ 2 x = 6 e^ 2 x (7+e^ 2 x )^2 d x & y 7+e^ 2 x = -3 7+e^ 2 x +C & y(0)=5 & 5 8 = -3 8 +C & C=1 & y=-3+7+e^ 2 x & y=e^ 2 x +4 & k=8 aligned

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