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JEE MainMathematicsIndefinite Integration

Let y = y(x) be the solution to the differential equation dy dx = ( ( x e )^ 2x - ( e x )^ 2x ) x for x > 0 , with the initial condition y(e) = 1 . The value of y(1) is equal to

Options

  1. Ae^4 - e^2 + 1 e^2
  2. Be^4 + 1 2e^2
  3. C-e^4 + 2e^2 + 1 2e^2
  4. De^2

Correct answer

B. e^4 + 1 2e^2

Step-by-step solution

Given the differential equation: dy dx = ( ( x e )^ 2x - ( e x )^ 2x ) x Integrating both sides with respect to x , we get: y(x) = ( ( x e )^ 2x - ( e x )^ 2x ) x , dx Let u = ( x e )^ 2x . Taking the natural logarithm on both sides: u = 2x ( x - 1) = 2x x - 2x Differentiating with respect to x : 1 u du dx = 2 x + 2x ( 1 x ) - 2 = 2 x du dx = 2u x Similarly, let v = ( e x )^ 2x . Taking the natural logarithm: v = 2x (1 - x) = 2x - 2x x Differentiating with respect to x : 1 v dv dx = 2 - 2 x - 2x ( 1 x ) = -2 x dv d

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