Abstract. In the paper, by virtue of the Faà di Bruno formula, some properties of the Bell polynomials of the second kind, and an inversion formula for the Stirling numbers of the first and second kinds, the authors establish meaningfully and significantly two identities which simplify coefficients in a family of ordinary differential equations associated with higher order Bernoulli numbers of the second kind.E-mail addresses: qifeng618@gmail.com, qifeng618@hotmail.com, qifeng618@qq.com, nnddww@gmail.com, bai.ni.guo@gmail.com, bai.ni.guo@hotmail.com. 2010 Mathematics Subject Classification. Primary 34A05; Secondary 11A25, 11B68, 11B73, 11B83. Key words and phrases. simplification; coefficient; ordinary differential equation; higher order Bernoulli number of the second kind; Stirling number of the first kind; Stirling number of the second kind; inversion formula; Bell polynomial of the second kind; Faà di Bruno formula.