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Question

If y1y=2x, then which of the following options is correct ?

A
(x2+1)d2ydx2+xdydx4y=0
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B
(x21)d2ydx2+xdydx4y=0
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C
(x2+1)d2ydx2+xdydx+4y=0
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D
(x2+1)d2ydx2xdydx4y=0
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Solution

The correct option is A (x2+1)d2ydx2+xdydx4y=0
y1y=2x
(y)22xy1=0
y=2x+4x2+42 (y>0)
y=(x+x2+1)2
dydx=2(x+x2+1)(1+xx2+1)
dydx=2(x+x2+1)2x2+1
dydx=2yx2+1

Squaring both the sides
(x2+1)(dydx)2=4y2

Differentiate w.r.t. x
(x2+1)×2(dydx)(d2ydx2)+2x(dydx)2=8ydydx

Dividing by 2dydx, we get
(x2+1)d2ydx2+xdydx=4y
(x2+1)d2ydx2+xdydx4y=0

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