Question Number 115000 by mnjuly1970 last updated on 22/Sep/20

            ....nice   math...     if  y =(cos(2x))^(−(1/2))  then  prove ::   y+y^(′′) = 3y^5                  ...m.n.july.1970...

Answered by Dwaipayan Shikari last updated on 22/Sep/20

y=(1/( (√(cos2x))))  y′=cos^(−(3/2)) 2x.sin2x  y′′=3cos^(−(5/2)) 2x.sin^2 2x+2cos^(−(1/2)) 2x  y+y′′=3(cos^(−(5/2)) 2x(1−cos^2 2x)+cos^(−(1/2)) 2x)  y+y′′=3cos^(−(5/2)) 2x−3cos^(−(1/2)) 2x+3cos^(−(1/2)) 2x  y+y′′=3y^5

Commented bymnjuly1970 last updated on 22/Sep/20

very nice very nice ...thank  you sir...

Commented byCanovas last updated on 29/Sep/20

Correct

Answered by $@y@m last updated on 23/Sep/20

y=(1/( (√(cos 2x))))     y^2 =sec2 x ...(1)  2yy′=2sec2xtan 2x  yy′=sec2xtan 2x  yy′=y^2 tan2 x  y′=ytan 2x ...(2)  y′′=2ysec ^2 2x+tan2 x.y′  y′′=2y.y^4 +tan 2x.ytan 2x  {using(1) &(2)  y′′=2y^5 +ytan ^2 2x  y′′=2y^5 +y(sec ^2 2x−1)  y′′=2y^5 +y(y^4 −1)  y+y′′=3y^5

Commented bymnjuly1970 last updated on 23/Sep/20

very nice awesom solution