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

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