4.6二阶非齐次常微分方程的求解

合集 - 数学物理方程 姜颖版 笔记(11)

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7.4.6二阶非齐次常微分方程的求解2024-11-14

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收起

二阶非齐次常微分方程 $$u^{″}(z)+p(z)u^{\prime }(z)+q(z)u(z)=f(z)$$

假设以上方程对应的齐次方程的线性独立的解为 $u_1(z)$，$u_2(z)$

$\{\begin{array}{l}{u}_1^{″}(z)+p(z)u_1^{\prime }(z)+q(z)u_1(z)=0\\ u_2^{″}(z)+p(z)u_2^{\prime }(z)+q(z)u_2(z)=0\end{array}$（齐次下方程解）

设该非齐次方程的特解为

$u_0(z)=y_1(z)u_1(z)+y_2(z)u_2(z)$

$u_0^{\prime }(z)=y_1^{\prime }(z)u_1(z)+y_1(z)u_1^{\prime }(z)+y_2^{\prime }(z)u_2(z)+y_2(z)u_2^{\prime }(z)$

$u_0^{\prime }(z)=y_1(z)u_1^{\prime }(z)+y_2(z)u_2^{\prime }(z)$ $$ [因为假设 $y_1^{\prime }(z)u_1(z)+y_2^{\prime }(z)u_2(z)=0$]

回代

$$y_1(z)[u_1^{″}(z)+p(z)u_1^{\prime }(z)+q(z)u_1(z)]+y_2(z)[u_2^{″}(z)+p(z)u_2^{\prime }(z)+q(z)u_2(z)]+y_1^{\prime }(z)u_1^{\prime }(z)+y_2^{\prime }(z)u_2^{\prime }(z)=f(z)$$

$\{\begin{array}{l}{y}_1^{\prime }(z)u_1(z)+y_2^{\prime }(z)u_2(z)=0\\ y_1^{\prime }(z)u_1^{\prime }(z)+y_2^{\prime }(z)u_2^{\prime }(z)=f(z)\end{array}$

可得：

$L(z)=\left|\begin{array}{cc}{u}_1(z)& u_2(z)\\ u_1^{\prime }(z)& u_2^{\prime }(z)\end{array}\right|\ne 0$ 即必有解

则

$$u(z)=a_1{u}_1(z)+a_2{u}_2(z)+u_0(z)=[a_1+y_1(z)]u_1(z)+[a_2+y_2(z)]u_2(z)$$