In this paper, we analyze the boundary value problem of a class of multi-order fractional differential equations involving the standard Caputo fractional derivative with the general periodic boundary conditions: \t\t\t{L(D)u(t)=f(t,u(t)),t∈[0,T],T>0,u(0)=u(T)>0,u′(0)=u′(T)>0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} L(D)u(t) = f(t,u(t)),\\quad t\\in[0,T], T>0, \\\\ u(0) = u(T)>0,\\qquad u'(0)=u'(T)>0, \\end{cases} $$\\end{document} where L(D)=sum^{n}_{i=0}a_{i}D^{S_{i}}, 1leq S_{0}<cdots<S_{n-1}<S_{n}<2, a_{i}inmathbb{R}, a_{n}neq0, and f:[0,T]timesmathbb{R}rightarrowmathbb{R} is a continuous operation. We get the Green’s function in terms of the Laplace transform. We obtain the existence and uniqueness of solution for the class of multi-order fractional differential equations. We investigate the blowing-up solutions to the special case f(t,u(t))=|u(t)|^{p}, a_{i}geq0, and give an upper bound on the blow-up time T_{mathrm{max}}.