Lineare gewöhnliche Differentialgleichung erster Ordnung
\( \begin{array}{l}z^{\prime}-2 z=\cos (x)-2 e^{3 x} \\ g(x)=-2 \\ s(x)=\cos (x)-2 e^{3 x} \\ \int g(x) d x=\int-2 d x=-2 x \\ \int s(x) e^{\int g(x) d x}=\int\left(\cos (x)-2 e^{3 x}\right) e^{-2 x} d x \\ =\int\left(\cos (x) e^{-2 x}-2 e^{x}\right) d x \\\end{array} \)
\( \begin{aligned} & =\frac{e^{-2 x}}{5}(\sin (x)-2 \cos (x))-2 e^{x} \\ \operatorname{allgemein}: z =e^{-\int \operatorname{g(x)} d x }\left[s(x) e^{\int \ g(x) d x} d x+c\right]\end{aligned} \)
\( z=\frac{\sin (x)}{5}-\frac{2}{5} \operatorname{cos}(x)-2 e^{3 x}+C e^{2 x} \)
