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MHT CET20242 May 2024Morning ShiftMathematicsDifferential EquationsActual

If x ~d y ~d x =y( y- x+1) , then general solution of this equation is

Options

  1. A( x y )= c y , where c is a constant of integration.
  2. B( x y )=c x , where c is a constant of integration.
  3. C( y x )= cy , where c is a constant of integration.
  4. D( y x )= c x , where c is a constant of integration:

Correct answer

D. ( y x )= c x , where c is a constant of integration:

Step-by-step solution

aligned & x ~d y ~d x =y( y- x+1) . & d y ~d x = y x [ ( y x )+1 ] & Put v = y x & y= v x & ~d y ~d x = v +x ~d v ~d x & From (i) aligned aligned & v +x ~d v ~d x = v ( v +1) & v +x ~d v ~d x = vlog v + v & x ~d v ~d x = v v & 1 v v d v= d x x aligned Integrating on both sides, we get 1 vlog v dv = d x x aligned & ( v )= x+ c ₁ & ( v )= x+ c where, c ₁= c & ( v )= (x c ) & v =x c & ( y x )= c x aligned

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