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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  			 		 $\lambda&gt;\frac {1}{n-1}$ 		and  			 		 $\mu&gt;0$ 		, 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  			 		 $r(y)&gt;yr^{\prime }(y)&gt;0$ 		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}&lt;\infty $ 		. We will also give a new proof of the existence of a constant  			 		 $y_{1}&gt;0$ 		such that  			 		 $r^{\prime \prime }(y_{1})=0$ 		,  			 		 $r^{\prime \prime }(y)&gt;0$ 		for any  			 		 $0&lt;y&lt;y_{1}$ 		, and  			 		 $r^{\prime \prime }(y)&lt;0$ 		for any  			 		 $y&gt;y_{1}$ 		.

Existence of hypercylinder expanders of the inverse mean curvature flow