In the note, the authors find several simple and explicit forms for a family of inhomogeneous linear ordinary differential equations studied in "D. Lim, Differential equations for Daehee polynomials and their applications, J. Nonlinear Sci. Appl. 10 (2017), no. 4, 1303-1315; Available online at http://dx.doi.org/10.22436/jnsa.010.04.02".

In the paper, the authors nd a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.

In the paper, by virtue of techniques in combinatorial analysis, the authors simplify two families of nonlinear ordinary differential equations in terms of the Stirling numbers of the first kind.