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The general solution of the differential equation x d y d x =y+x ( y x ) is

Options

  1. A( x y )=C x
  2. B( x y )=C y
  3. C( y x )=C x
  4. D( y x )= C x

Correct answer

C. ( y x )=C x

Step-by-step solution

The given differential equation is x dy dx = y + x ( y x ) . Dividing by x , we get dy dx = y x + ( y x ) . This is a homogeneous differential equation. Let y = vx , then dy dx = v + x dv dx . Substituting these into the equation: v + x dv dx = v + (v) x dv dx = (v) Separating the variables: dv (v) = dx x (v) dv = dx x Integrating both sides: (v) dv = dx x | (v)| = |x| + |C| | (v)| = |Cx| (v) = Cx Substituting v = y x back into the equation: ( y x ) = Cx Answer: ( y x )=C x

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