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Manipal MET2020MathematicsDifferential Equations

The general solution of the differential equation d y d y =y x-y^2 x is

Options

  1. Ax=(C+ x) y
  2. By=(C+ y) x
  3. Cx=(C+ x) y
  4. Dy=(C+ x) x

Correct answer

C. x=(C+ x) y

Step-by-step solution

Given differential equation is d y d x =y x-y^2 x d y d x -y x=-y^2 x 1 y^2 d y d x - x y =- x ....(i) Put -1 y =u 1 y^2 d y d x = d u d x From Eq. (i), d u d x + x u=- x ....(ii) This is linear differential equation of the form d u d x +P u=Q , where P= x, Q=- x FF = e ^ x d x = e ^ x = x Hence, general solution is aligned & & u IF = IF Q d x+C₁ & & u x= ( x) (- x) d x+C₁ & & u x=- ^2 x d x+C₁ & & u x=- x+C₁ aligned - x y =- x+C₁ [ u= -1 y ] array ll & x=y ( x-C₁ ) & x=y( x+C) [ where, C=-C₁ ] array

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