二阶常系数非齐次线性微分方程特解的直接积分法

A direct integration method to get particular solutions for second order linear differential equations with constant coefficients

为了更简便地求出二阶常系数线性非齐次微分方程的一个特解,给出了一种直接积分方法.若已知二阶方程y″+py′+qy=f(x)的一个实特征根λ,可以使用直接积分的方法得到非齐次方程的一个特解y*=exp(-(λ+p)x)∫[(exp((2λ+p)x∫)α(x)dx)dx].当方程有2个相等实特征根时,特解的表示形式更加简洁.更主要的是,该直接积分法除了适用于教材中两种特殊类型函数f(x)的非齐次方程,也可用于任意函数f(x)的非齐次方程.

An integration method is proposed to get a particular solution for second order linear differential equations,with constant coefficients.If the equation y″＋py′＋qy=f（x） has two real characteristic roots,and one of them is given as λ,a particular solution can be got from twice integration directly and repressed as y＊=exp（-（λ＋p）x）〗.When two roots are equal,the particular solution has more concise form.The most important is that the method given in this article can be used in equations with different inhomogeneous terms,involving the two types of functions explained in the text book,and other nonlinear functions.