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We will give a new proof of the existence of non-compact homothetic solitonsof the inverse mean curvature flow (cf. \cite{DLW}) 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 $\frac{1}{n}<\lambda<\frac{1}{n-1}$ and $\mu<0$, we willgive a new proof of the existence of a unique solution $r(y)\inC^2(\mu,\infty)\cap C([\mu,\infty))$ of the equation$\frac{r_{yy}(y)}{1+r_y(y)^2}=\frac{n-1}{r(y)}-\frac{1+r_y(y)^2}{\lambda(r(y)-yr_y(y))}$,$r(y)>0$, in $(\mu,\infty)$ which satisfies $r(\mu)=0$ and$r_y(\mu)=\lim_{y\searrow\mu}r_y(y)=+\infty$. We also prove that there existconstants $y_2>y_1>0$ such that $r_y(y)>0$ for any $\mu<y<y_1$, $r_y(y_1)=0$,$r_y(y)<0$ for any $y>y_1$, $r_{yy}(y)<0$ for any $\mu<y<y_2$, $r_{yy}(y_2)=0$and $r_{yy}(y)>0$ for any $y>y_2$. Moreover $\lim_{y\to +\infty}r(y)=0$ and$\lim_{y\to +\infty}yr_y(y)=0$.

Another proof of the existence of homothetic solitons of the inverse mean curvature flow