If y = etan x  , prove that  (cos2 x)d2y/dx2 – ( 1 + sin 2x).dy/dx = 0

Explain in great detail

Asked by haroonrashidgkp | 25th Aug, 2018, 07:05: PM

Expert Answer:

begin mathsize 16px style straight y equals straight e to the power of tanx
dy over dx equals straight e to the power of tanx cross times sec squared straight x
dy over dx equals ysec squared straight x......... left parenthesis straight i right parenthesis
fraction numerator straight d squared straight y over denominator dx squared end fraction equals 2 ysecxsecxtanx plus sec squared straight x dy over dx
fraction numerator straight d squared straight y over denominator dx squared end fraction equals 2 ysec squared xtanx plus sec squared straight x dy over dx
cos squared straight x fraction numerator straight d squared straight y over denominator dx squared end fraction equals 2 ytanx plus dy over dx
Consider comma
LHS space
equals space cos squared straight x fraction numerator straight d squared straight y over denominator dx squared end fraction minus left parenthesis 1 plus sin 2 straight x right parenthesis dy over dx
equals 2 ytanx plus dy over dx minus dy over dx minus sin 2 straight x dy over dx
equals 2 ytanx minus sin 2 straight x dy over dx
equals 2 ytanx minus sin 2 straight x cross times ysec squared straight x....... from space left parenthesis straight i right parenthesis
equals 2 straight y cross times sinx over cosx minus 2 ysinxcosx cross times fraction numerator 1 over denominator cos squared straight x end fraction
equals 2 straight y cross times sinx over cosx minus 2 straight y cross times sinx over cosx
equals 0 end style

Answered by Sneha shidid | 28th Aug, 2018, 10:54: AM