二阶常系数非齐次微分方程的特解怎么设,有什么规律

2025年03月15日 04:38
有1个网友回答
网友(1):

较常用的几个:


1、Ay''+By'+Cy=e^mx


特解    y=C(x)e^mx


2、Ay''+By'+Cy=a sinx + bcosx


特解    y=msinx+nsinx


3、Ay''+By'+Cy= mx+n              


特解    y=ax


二阶常系数线性微分方程是形如y''+py'+qy=f(x)的微分方程,其中p,q是实常数。自由项f(x)为定义在区间I上的连续函数,即y''+py'+qy=0时,称为二阶常系数齐次线性微分方程。

扩展资料

F″(λ)/2!z″+F′(λ)/1!z′+F(λ)z=pm(x) ,这里F(λ)=λ^2+pλ+q为方程对应齐次方程的特征多项式。


升阶法:


设y''+p(x)y'+q(x)y=f(x),当f(x)为多项式时,设f(x)=a0x^n+a1x^(n-1)+…+a(n-1)x+an,此时,方程两边同时对x求导n次,得


y'''+p(x)y''+q(x)y'=a0x^n+a1x^(n-1)+…+a(n-1)x+an……


y^(n+1)+py^(n)+qy^(n-1)=a0n!x+a1(n-1)!


y^(n+2)+py^(n+1)+qy^(n)=a0n!


令y^n=a0n!/q(q≠0),此时,y^(n+2)=y^(n+1)=0。由y^(n+1)与y^n通过倒数第二个方程可得y^(n-1),依次升阶,一直推到方程y''+p(x)y'+q(x)y=f(x),可得到方程的一个特解y(x)。

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