y=xx .prove that d2y/dx​2-1/y*(dy/dx)2​-y/x=0

Dear student
We have y=xxDifferentiate it w.r.t.x., we getdydx=ddxxxdydx=ddxelog(x)xdydx=exlog(x)×ddxx logxdydx=exlogxlogx+1dydx=xxlogx+1    ...(1)Differentiate it w.r.t.x., we getd2ydx2=ddxxxlogx+1+xx×ddxlogx+1d2ydx2=xxlogx+12+xx1x   using 1Considerd2ydx2-1ydydx2-yx=xxlogx+12+xx1x-1xxxxlogx+12-xxx=xxlogx+12-x2xxxlogx+12=xxlogx+12-xxlogx+12=0Hence, d2ydx2-1ydydx2-yx=0
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