关于广义Riccati方程的可积条件——与赵临龙先生商榷

On Integrability Conditions of Generalized Riccati Equation——To discuss about it with Mr.Zhao Linlong

指出《科学通报》1998年第1期《Riccati微分方程一个新的可积条件》中的错误并给出正确结论,即方程y′=P(x)yn+Q(x)y+R(x)的可积条件不是R=K′Pe∫n(Q-βD)dx(K′,β为常数),而是Q-1nRR′-PP′nPRn-1=τ(τ为常数);给出了满足这一条件的方程的通积分;推广了该方程原有的可积条件R=KPe∫nQdx(K为常数).

An error in Zhao Linlong＇s ＂A New Integrability Condition for Riccati Differential Equation＂ in Chinese Science Bulletin, 1998,43：439-440, is pointed and corrected. That is, integrability condition of equation y＇=P（x）y^n＋Q（x）y＋R（x） is notR=K＇Pe^n∫ （Q-βD）dx （K＇,β are constants） but [Q-1-n（R＇/R-P＇/P）]^n/PR^n-1=r （r is a constant）. Its general integral that satisfies this condition is given too, Its original integrability condition R=KPe^n∫Qdx （k is a constant） is extended really.