If `ysqrt(x^2+1)=log(sqrt(x^2+)1-x),`show that`" "` `(x^2+1)(dy)/(dx)+x y+1=0`

Explanation

`sqrt{x^{2}+1}=log {sqrt{x^{2}+1}-x}`


Differentiating with reset to be `x` we get


`sqrt{x^{2}+1} frac{d y}{d x}+frac{1}{2 sqrt{x^{2}+1}} cdot 2 x cdot y=frac{1}{sqrt{x^{2}+1-x}}{frac{1}{2 sqrt{x^{2}+1}} 2 {x}-1}`


`sqrt{x^{2}} frac{d x}{d x}+frac{x}{sqrt{x^{2}+1}}=frac{1}{sqrt{x^{2}+1-x}}{frac{x}{sqrt{x^{2}+1-1}}}`


`Rightarrow sqrt{x^{2}+1} frac{d y}{d i x}+frac{x y}{sqrt{x^{2}+1}}=frac{1}{sqrt{x^{2}+1}-x}{frac{x^{-} sqrt{x^{2}+1}}{sqrt{x^{2}+1}}}`


`Rightarrow sqrt{x^{2}+1} frac{d y}{d x}+frac{x y}{sqrt{x^{2}+1}}=frac{-1}{sqrt{x^{2}+1}}`


`Rightarrow(x^{2}+1) frac{d y}{d x}+x y=-1`


`Rightarrow(x^{2}+1) frac{d y}{d x}+x y+1=0 .`