if y=x+1/y prove that (x²-y²+3)dy/dx=1

Dear Aritra,

Solution:-  y = x +1/y.................(1)

              dy/dx = 1 - (1/y2) (dy/dx)

           [1 + (1/y2)] (dy/dx) = 1..................(2)

        From eq. (1), y-1/y = x. On squaring both sides, we get-

 (y-1/y )2= x2 or  y2 + 1/y2 -2= x2 or 1/y2 = (x2 - y2+2)

Putting the value of 1/y2  in eq. (2), we get-

  [1 + (x2 - y2+2)] (dy/dx) = 1

(x2 - y2+3) (dy/dx) = 1

Hence proved.                       

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x=y-1/y ...........1

differentiating wrt x

 1 = dy/dx -1/y² (dy/dx)

dy/dx=y² /1+y²  ................2

we have to prove (x² -y² +3 )dy/dx =1       or

                      x² -y² +3 =1/(dy/dx)

taking LHS

                    = (x-y)(x+y)+3                     (putting x from eq 1)

                   = y² +1/y²  .....................4

taking RHS

                     =1/(dy/dx)=(y²+1)/y² .............5

FROM 4 & 5

LHS = RHS

hence proved