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JEE Main202529 Jan 2025Morning ShiftMathematicsDifferential EquationsActual

Let (y=y(x) ) be the solution of the differential equation ( x ( _ e ( x) )^2 dy + ( x-3 y x _ e ( x) ) d x=0, x (0, 2 ) ). If (y ( 4 )= -1 _ e 2 ), then (y ( 6 ) ) is equal to :

Options

  1. A( 1 _e(3)- _e(4) )
  2. B( 2 _e(3)- _e(4) )
  3. C( 1 _e(4)- _e(3) )
  4. D(- 1 _e(4) )

Correct answer

A. ( 1 _e(3)- _e(4) )

Step-by-step solution

aligned & d y d x - 3 x x x y=- x x( x)^2 & I.F. =e^ - 3 x x d x & Let x=t & - x d x=d t & e^ 3 ^ d t t =e^ 3 t =t^3=( x)^3 & Solution will be & y( x)^3=- ( x)( x) d x & y( x)^3= ( x)^2 2 +c & y ( 4 )=- 1 2 & c=0 & y= 1 2( x) & y ( 6 )= 1 2 1 ( 6 ) & = 1 2 [ 1 ( 3 2 ) ] & = 1 2 1 3 - 2 & = 1 3- 4 aligned

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