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NTA Abhyas JEE Main2020MathematicsDifferential EquationsPractice

The general solution of the differential equation 2 x y – x d y + y d x = 0 is (Here x, y > 0)

Options

  1. Alog ⁡ x + y x = c
  2. Blog ⁡ y – x y = c
  3. Clog ⁡ y + x y = c
  4. DNone of these

Correct answer

C. log ⁡ y + x y = c

Step-by-step solution

We have, d y d x = y x - 2 x y which is homogeneous. Put, y = V x so that d y d x = x d V d x + V ⇒ x d V d x = V 1 - 2 V - V = 2 V 3 / 2 1 - 2 V ⇒ d x x = 1 - 2 V 2 V 3 / 2 d V = 1 2 V 3 / 2 - 1 V d V Integrating, we get, - c + log ⁡ x = - V - 1 / 2 - log ⁡ V = - x y - log ⁡ y + log ⁡ x ⇒ log ⁡ y + x y = c .

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