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JEE Main20241 Feb 2024Evening ShiftMathematicsDifferential EquationsActual

If d x d y = 1 + x − y 2 y , x 1 = 1 , then 5 x 2 is equal to:

Correct answer

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Step-by-step solution

Given: d x d y = 1 + x - y 2 y ⇒ d x d y = 1 + x y - y ⇒ d x d y - 1 + x y = - y Now let 1 + x = t ⇒ d x d y = d t d y ⇒ d t d y - t y = - y ⇒ IF = e ∫ - 1 y d y ⇒ IF = e - log y ⇒ IF = e log 1 y ⇒ IF = 1 y So, the solution of the given differential equation is t y = ∫ 1 y × - y d y ⇒ 1 + x y = ∫ 1 y × - y d y ⇒ 1 + x y = - y + c Also, x 1 = 1 ⇒ 2 1 = - 1 + c ⇒ c = 3 ⇒ 1 + x y = - y + 3 Putting y = 2 ⇒ 1 + x 2 = - 2 + 3 ⇒ 1 + x = 2 ⇒ x = 1 ⇒ 5 x = 5

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