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if y=x+1/y prove that (x 2 -y 2 +3)dy/dx=1

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

Grade:10

2 Answers

Priyansh Bajaj AskiitiansExpert-IITD
30 Points
11 years ago

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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Priyansh Bajaj

vikas askiitian expert
509 Points
11 years ago

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

 

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