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MHT CET20228 Aug 2022Morning ShiftMathematicsDifferential EquationsActual

The differential equation y^ = y x+ x y has general solution given by (Where C is a constant of integration.)

Options

  1. Ay=C e^ 2 x y
  2. By=C e^ 2 ( y x )
  3. Cy=C e^ 2 ( x y )
  4. Dy=C e^ -2 x y

Correct answer

C. y=C e^ 2 ( x y )

Step-by-step solution

aligned & d y ~d x = y x+ x y let y=v x & d y ~d x =v+x d v ~d x & v+x d v ~d x = v x x+ x v x = v 1+ v & x d v ~d x = v 1+ v -v= -v v 1+ v & - 1+ v v v ~d v= d x x & - (v^ -3 2 +v⁻¹ ) d v= d x x & 2 v -1 2 - v= x+ c^ & 2 x y - y x = c^ x aligned aligned & 2 x y = (c^ x y x ) & c e^ 2 x y aligned

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