常系数非齐次微分方程初值问题的显式解

AN EXPLICIT SOLUTION FOR INITIAL PROBLEM OF NON HOMOGENEOUS DIFFERENTIAL EQUATION WITH CONSTANT COEFFICIENTS

将一般常系数非齐次线性微分方程初值问题的解直接表示成其系数与初始值的函数，从而避免了按通常解法带来的求解相应高次代数方程的麻烦。

We consider the initial problem of the following non homogeneous differential equation with constant coefficients( A ) y (n) (x)-∑n-1i=0α iy (i) (x)=∑∞j=0b jx j y (i) (x) x=0=C i (i=0,1,…,n-1)where a i and c i (i=0,1,…,n-1) and b j (j=0,1,…) are constants, n≥1 . Its general solution is given by the following formulay(x)=∑n-1k=0c kk!x k+∑∞k=n{∑n-1i=0{∑ij=0{F(k-n-j)}α i-j }c i+ ∑k-nj=0{j!b jF(k-n-j)}}x kk!in whichF(m)=∑np 0+(n-1)p 1+…+p n-1 =m(p 0+p 1+…+p n-1 )!p 0!p 1!…p n-1 !α p 0 0α p 1 1…α p n-1 n-1 , for m>0 1, for m=0 0, for m<0