In this paper, the author considers a PDE of the form \[ S_k(D^2v)+\alpha v+\beta \xi\cdot\nabla v=0, \] where \(v>0\) on \(\mathbb{R}^{n}\) and \(v(0)=a>0\). In the above equation the operator \(S_k\) is the \(k\)-Hessian; moreover, \(D^2v\) is the Hessian of \(v\). The \(k\)-Hessian of \(v\) is defined by \[ S_k(D^2v):=\sum_{1\le i_10\) on \((0,+\infty)\), and satisfying the initial data \[ v(0)=a\text{ and }v'(0)=0. \] The principal result of the paper asserts the existence of a unique solution of the preceding ODE under the assumptions that \(1\le k0\). Moreover, it is proved that if \(\alpha\neq0\) and \(\delta:=\frac{\beta}{\alpha}\), then the function \[ E(r):=r^2\big(v(r)\big)^{2\delta} \] is strictly monotone increasing for \(r>0\). All in all, the paper seems to be well written and adequately motivated. In particular, this would be of interest for researchers interested in ordinary differential equations, especially inasmuch as how they arise from radially symmetric solutions of PDEs.