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AP EAMCET2010MathematicsDifferential Equations

A family of curves has the differential equation x y d y d x =2 y^2-x^2 . Then, the family of curves is

Options

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

Correct answer

D. y^2=x^2+c x^4

Step-by-step solution

x y d y d x =2 y^2-x^2 d y d x = 2 y x - x y d y d x - 2 y x = -x y y d y d x -2 y^2 x =-x (i) Put, v=y^2 (ii) ( d v d x =2 y d y d x ) ( 1 2 d v d x =y d y d x ) From Eq. (i) 1 2 d v d x - 2 v x =-x d v d x - 4 v x =-2 x IF =e^ p d x =e^ - 4 x d x =e^ -4 x =e^ x⁻⁴ =x⁻⁴ Complete solution is x⁻⁴ v= (-2 x) x⁻⁴ d x+c v x^4 =-2 d x x^3 +c v x^4 = 1 x^2 +cv=x^2+c x^4y^2=c x^4+x^2 [ from Eq. (ii) ] which is the required family of curves.

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