适用于 初等可积微分方程の全解 > 二阶常系数非齐次线性微分方程

设对应齐次方程的通解为

$$ y=C_{1}\varphi_{1}(x)+C_{2}\varphi_{2}(x) $$

可以假设齐次方程的常数变易为函数，构成原非齐次方程的通解

$$ y=C_{1}(x)\varphi_{1}(x)+C_{2}(x)\varphi_{2}(x) $$

不妨规定

$$ C_{1}’(x)\varphi_{1}(x)+C_{2}’(x)\varphi_{2}(x)=0 $$

则

$$ \left{ \begin{align} y’ & =C_{1}(x)\varphi_{1}’(x)+C_{2}(x)\varphi_{2}’(x) \ y’’ & =C_{1}(x)\varphi_{1}’’(x)+C_{2}(x)\varphi_{2}’’(x)+C_{1}’(x)\varphi_{1}’(x)+C_{2}’(x)\varphi_{2}’(x) \end{align} \right. $$

带入原方程化简

$$ \left{ \begin{align} C_{1}’(x)\varphi_{1}(x)+C_{2}’(x)\varphi_{2}(x) & =0 \ C_{1}(x)’\varphi_{1}’(x)+C_{2}’(x)\varphi_{2}’(x) & =f(x) \end{align} \right. $$

可以解得 $C_{1}(x)$ 和 $C_{2}(x)$ 最后解得通解