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The solution of the differential equation x d y d x = y ln ⁡ y 2 x 2 is (where, c is an arbitrary constant)

Options

  1. Ay = x . e c x + 1
  2. By = x . e c x - 1
  3. Cy = x 2 . e c x + 1
  4. Dy = x . e c x 2 + 1 2

Correct answer

D. y = x . e c x 2 + 1 2

Step-by-step solution

Putting y = x v   and   d y d x = v + x d v d x So, v + x d v d x = 2 v ln ⁡ v ⇒   ∫ d x x = ∫ d v v ( 2 ln ⁡ v - 1 ) On integrating, we get,   y = x ⋅ e c x 2 +   1 2

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