二阶方程常数变易法

公式推导

已知方程：

\[\begin{cases} y^{\prime \prime}(x) + \omega^{2}(x) = f(x)\\ y(0) = \alpha, \quad y^{\prime}(0) = \beta \end{cases}\]

可以得到相应的齐次方程：$ y^{\prime \prime}(x) + \omega^{2}(x) = 0 $

通解：$ y(x) = C_{1}cos(\omega x) + C_{2}sin(\omega x) $

由常数变易法，$ y(x) = u(x)cos(\omega x) + v(x)sin(\omega x) $

构造二元线性方程组

\[y^{\prime}(x) = [u^{\prime}(x) cos(\omega x) + v^{\prime}(x) sin(\omega x) + [-\omega u(x) sin(\omega x) + \omega v(x)cos(\omega x)]\]

令第一个方程为 0，即

\[\begin{equation} u^{\prime}(x) cos(\omega x) + v^{\prime}(x) sin(\omega x) = 0 \tag{1} \end{equation}\]

余下部分再求导：

\[y^{\prime \prime}(x) = [-\omega u^{\prime}(x) sin(\omega x) + \omega v^{\prime}(x) cos(\omega x)] - [\omega^{2} u(x) cos(\omega x) + \omega^{2} v(x) sin(\omega x)]\]

代入原方程得到：

\[\begin{equation} -\omega u^{\prime}(x) sin(\omega x) + \omega v^{\prime}(x) cos(\omega x) = f(x) \tag{2} \end{equation}\]

(1)、(2)联立得

\[\begin{bmatrix} cos(\omega x) & sin(\omega x) \\ -\omega sin(\omega x) & \omega cos(\omega x) \end{bmatrix} \begin{bmatrix} u^{\prime}(x) \\ v^{\prime}(x) \end{bmatrix} =\begin{bmatrix} 0 \\ f(x) \end{bmatrix}\]

由克莱姆法则：

\[D = \begin{vmatrix} cos(\omega x) & sin(\omega x) \\ -\omega sin(\omega x) & \omega cos(\omega x) \end{vmatrix} = \omega\] \[D_{1} = \begin{vmatrix} 0 & sin(\omega x) \\ f(x) & \omega cos(\omega x) \end{vmatrix} = -f(x)sin(\omega x)\] \[D_{2} = \begin{vmatrix} cos(\omega x) & 0 \\ -\omega sin(\omega x) & f(x) \end{vmatrix} = f(x)cos(\omega x)\]

解得：

\[u'(x) = \frac{D_1}{D} = -\frac{1}{\omega}f(x)sin(\omega x)\] \[v'(x) = \frac{D_2}{D} = \frac{1}{\omega}f(x)cos(\omega x)\]

代入通解中得到：

\[\begin{equation} y = -\frac{1}{\omega} cos(\omega x) \int_{0}^{x} sin(\omega \xi) f(\xi) d\xi + \frac{1}{\omega} \int_{0}^{x} cos(\omega \xi)f(\xi)d\xi + C_3 cos(\omega x) + C_4 sin(\omega x) \tag{3} \end{equation}\]

利用初始条件：$ C_3 = \alpha, C_4 = \frac{\beta}{\omega} $ 代入 (3) 中，并用和差化积公式化简：

\[y(x) = \alpha cos(\omega x) + \frac{\beta}{\omega} sin(\omega x) + \frac{1}{\omega} \int_{0}^{x}sin[\omega (x - \xi)] f(\xi)d\xi\]

补充知识——和差化积、积化和差公式

和差化积

\[sin \alpha + sin \beta = 2 sin \frac{\alpha + \beta}{2} cos \frac{\alpha - \beta}{2}\] \[sin \alpha - sin \beta = 2 cos \frac{\alpha + \beta}{2} sin \frac{\alpha - \beta}{2}\] \[cos \alpha + cos \beta = 2 cos \frac{\alpha + \beta}{2} cos \frac{\alpha - \beta}{2}\] \[cos \alpha - cos \beta = - 2 sin \frac{\alpha + \beta}{2} sin \frac{\alpha - \beta}{2}\]

积化和差

\[sin \alpha cos \beta = \frac{1}{2} [sin(\alpha + \beta) + sin(\alpha - \beta)]\] \[cos \alpha sin \beta = \frac{1}{2} [sin(\alpha + \beta) - sin(\alpha - \beta)]\] \[cos \alpha cos \beta = \frac{1}{2} [cos(\alpha + \beta) + cos(\alpha - \beta)]\] \[sin \alpha sin \beta = -\frac{1}{2} [cos(\alpha + \beta) - cos(\alpha - \beta)]\]