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Question

If y=log(x+x2+a2) then show that (x2+a2)d2ydx2+xdydx=0.

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Solution

Given y=log(x+x2+a2)ey=x+x2+a2
Differentiate both sides w.r.t x
ey(dydx)=1+2x2x2+a2
eydydx=eyx2+a2dydx=1x2+a2
Again differentiate on both sides w.r.t x
d2ydx2=2x2(x2+a2)3/2
(x2+a2)d2ydx2=xx2+a2
(x2+a2)d2ydx2=xdydx
(x2+a2)d2ydx2+xdydx=0

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