y=xx prove that d2y/dx2-1/y*(dy/dx)2-y/x=0 - Maths - Continuity and Differentiability

Question

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

Neha Sethi · Accepted Answer

Dear student
We have y=xxDifferentiate it w r t x
, we getdydx=ddxxx⇒dydx=ddxelog(x)x⇒dydx=exlog(x)×ddxx logx⇒dydx=exlogxlogx+1⇒dydx=xxlogx+1    
(1)Differentiate it w r t x
, we getd2ydx2=ddxxxlogx+1+xx×ddxlogx+1⇒d2ydx2=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
Regards

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

y=x^x .prove that d²y/dx²-1/y*(dy/dx)²-y/x=0