If the solution of the differential equation d y d x = x 3 + x y 2 y 3 - y x 2 is y k - x k = 2 x 2 y 2 + λ (where, λ is an arbitrary constant), then the value of k is

Question

If the solution of the differential equation d y d x = x 3 + x y 2 y 3 - y x 2 is y k - x k = 2 x 2 y 2 + λ (where, λ is an arbitrary constant), then the value of k is, 2, 4, 1, 3 2

MARKS App · Accepted Answer

y d y x d x = x 2 + y 2 y 2 - x 2 Let, y 2 = Y ; x 2 = X ⇒ y d y x d x = d Y d X Hence, the equation is d Y d X = X + Y Y - X ⇒ Y d Y - X d X = X d Y + Y d X On integrating we get Y 2 2 - X 2 2 = ∫ d X Y = X Y + c or Y 2 - X 2 = 2 X Y + λ ⇒ y 4 - x 4 = 2 x 2 y 2 + λ

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