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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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Regards,Askiitians ExpertsPriyansh Bajaj
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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