線型斉次形の一般解と線型非斉次形の特解の解の和は線型非斉次形の一般解になる

$ \forall y,\tilde y:\Complex\to\Complex\forall A\in\Complex;

$ \begin{dcases}\frac{\mathrm dy}{\mathrm dx}+f(x)y&=0\\\frac{\mathrm d\tilde y}{\mathrm dx}+f(x)\tilde y&=g(x)\end{dcases}

$ \implies\frac{\mathrm d}{\mathrm d x}(Ay+\tilde y)+f(x)(Ay+\tilde y)=A\frac{\mathrm dy}{\mathrm dx}+Af(x)y+\frac{\mathrm d\tilde y}{\mathrm dx}+f(x)\tilde y

$ =0+g(x)

$ \underline{\iff\frac{\mathrm d}{\mathrm d x}(Ay+\tilde y)+f(x)(Ay+\tilde y)=g(x)\quad}_\blacksquare

一般的な線型微分方程式での証明