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If y=(ax+b)/(cx+d) and if (a+d)=0 then show that (y-x)y"=2y(1+y')

If y=(ax+b)/(cx+d) and if (a+d)=0 then show that (y-x)y"=2y(1+y')

Grade:12th pass

1 Answers

Giridharan
13 Points
3 years ago
y=(ax+b)/(cx+d) and a+d=0
a+d=0 ⇒d=-a
Hence y=(ax+b)/(cx-a)
⇒y(cx-a)=(ax+b)
Differentiating w.r.t.x
cy+(cx-a)y1= a––––(1)
Differentiating again w.r.t.x
2cy1+(cx-a)y2=0–––(2)
⇒a = 2cy1/y2 + cx–––(3)
(1) ⇒ cy +(cx-a)y1 = a
Using the value of a from (3) and sub in (1) and eliminating fractions
we get 2y1(1+y1)=y2(y-x)
 

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