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JEE Main202330 Jan 2023Morning ShiftMathematicsDifferential EquationsActual

Let the solution curve y = y ( x ) of the differential equation d y d x - 3 x 5 tan - 1 x 3 1 + x 6 3 2 y = 2 x exp x 3 - tan - 1 x 3 ( 1 + x ) 6 pass through the origin. Then y ( 1 ) is equal to:

Options

  1. Aexp 4 - π 4 2
  2. Bexp π - 4 4 2
  3. Cexp 1 - π 4 2
  4. Dexp 4 + π 4 2

Correct answer

A. exp 4 - π 4 2

Step-by-step solution

Given differential equation is d y d x + - 3 x 5 tan - 1 x 3 1 + x 6 3 / 2 y = 2 x e x 3 - tan - 1 x 3 1 + x 6 This is a linear differential equation of the form d y d x + P y = Q . Hence, I . F . = e ∫ - 3 x 5 tan - 1 x 3 1 + x 6 3 / 2 d x Put tan - 1 x 3 = u ⇒ 3 x 2 1 + x 6 d x = d u I . F . = e - ∫ u tan u 1 + tan 2 u d u ⇒ I . F . = e - ∫ u tan u sec u d u ⇒ I . F . = e - ∫ u sin u d u ⇒ I . F . = e - - u cos u + ∫ cos u d u ⇒ I . F . = e u cos u - sin

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