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

Question

The general solution of the differential equation 2 x y – x d y + y d x = 0 is (Here x, y > 0), log ⁡ x + y x = c, log ⁡ y – x y = c, log ⁡ y + x y = c, None of these

MARKS App · Accepted Answer

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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