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We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in 
$\mathbb {R}^{n}\times \mathbb {R}$, 
$n\ge 2$, of the form 
$(r,y(r))$ or 
$(r(y),y)$, where 
$r=|x|$, 
$x\in \mathbb {R}^{n}$, is the radially symmetric coordinate and 
$y\in \mathbb {R}$. More precisely, for any 
$\lambda>\frac {1}{n-1}$ and 
$\mu>0$, we will give a new proof of the existence of a unique even solution 
$r(y)$ of the equation 
$\frac {r^{\prime \prime }(y)}{1+r^{\prime }(y)^{2}}=\frac {n-1}{r(y)}-\frac {1+r^{\prime }(y)^{2}}{\lambda (r(y)-yr^{\prime }(y))}$ in 
$\mathbb {R}$ which satisfies 
$r(0)=\mu $, 
$r^{\prime }(0)=0$ and 
$r(y)>yr^{\prime }(y)>0$ for any 
$y\in \mathbb {R}$. We will prove that 
$\lim _{y\to \infty }r(y)=\infty $ and 
$a_{1}:=\lim _{y\to \infty }r^{\prime }(y)$ exists with 
$0\le a_{1}<\infty $. We will also give a new proof of the existence of a constant 
$y_{1}>0$ such that 
$r^{\prime \prime }(y_{1})=0$, 
$r^{\prime \prime }(y)>0$ for any 
$0<y<y_{1}$, and 
$r^{\prime \prime }(y)<0$ for any 
$y>y_{1}$.
