一类常微分方程的解析解

The Analytic Solution of a Class of Differential Equation

设Q=Q(x)∈C,F=F(x),G=G(x)∈C',G(x)≠0,μ,A_μ,A_I(i=0,1,2,…,n-1)均为实常数,则方程 ?? 可积具有通解■一阶线性方程、Bemoulli方程是此定理的特例.文中并对Riccati方程和Appel方程的可积条件及通解形式进行了讨论.它们的可积性结果也是本定理的特例.

Given:Q=Q(x)∈C,F=F(x), G = G(x)∈C', G (x)≠0 ,μ , α_μ , α_i (i=0, 1 ,2 ,…, n-1) are real constants .Then equation: Y'-(G '/G)y=Q [a_μ(y+F)~μ +■(y+F)~i]+(G'/G)F-F' is integrable and possesses general solution ■ First order linear equation and Bernoulli equation are the particular cases of this theorem . ■ this paper has a discussed the integrable condition and the general solution forms of the Rio??ati equation and the Appel equation , Their integrability results are also the particular cases of this theorem .