二阶常系数非齐次线性微分方程特解

Special Solutions of Second Order Constant Coefficient Non-homogeneous Linear Differential Equations

设方程y″+py′+qy=e^(λx )P_(m)(x)有特解y*=R(x)eλx,P_(m)(x)为m次多项式.当λ是特征方程单根时,R′(x)是m次多项式,R(x)是m+1次多项式.对比R(x)=(ax+b)R_(m)(x)和R(x)=xR_(m)(x)两种假设(R_(m)(x)为m次多项式),对后者的合理性进行分析,并证明按后者所得特解是唯一存在的.

Let y*=R(x)e^(λx )be the special solution of y″+py′+qy=e^(λx )P_(m)(x),P_(m)(x)is an m-degree polynomial.Ifλis the single root of characteristic equation,then R′(x)is an m-degree polynomial and R(x)is an m+1-degree polynomial.The two hypotheses R(x)=(ax+b)R_(m)(x)and R(x)=xR_(m)(x)are compared(R_(m)(x)is an m-degree polynomial),rationality of the latter is analyzed,and the special solution corresponds to the latter is proved to be unique.