If y=log (rootx + 1/rootx)^2, then prove x (x+1)^2 y2 + (x+1)^2 y1 = 2

Question

Y2  means d^2y/(dx)^2

Y1  means dy/dx

Answer 1

y=log(√x+ 1/√x)²=log(x+ 1/x+2)=log((x+1)²/x²)

 

y'=x/(x+1)²(2(x+1)x-(x+1)²)/x²=(x-1)/((x+1)x)

y''=(x(x+1)+(1-x)(2x+1))/(x²(x+1)²)=1/x²-2/(1+x)²

 

 

x(x+1)²(1/x²-2/(1+x)²)+(x+1)²(x-1)/((x+1)x)=

=(x+1)²/x-2x+(x²-1)/x=(x²+2x+1+x²-1)/x-2x=2x+2-2x=2.