\( y^{(5)}-y^{(4)}+8 y^{(3)}-8 y^{''}+16 y^{\prime}-16 y=2 e^{3 x} \)
\( 5 x^{5}-k^{4}+8 x^{3}-8 k^{2}+16 k-16=0 \)
\( (k-1)\left(k^{2}+4\right)^{2}=0 \)
\( \Rightarrow \quad k-1=0 \)
$$ k_{1}=0 \Rightarrow y_{1}-c_{1} e x $$
\( \Rightarrow\left(k^{2}+4\right)^{2}=0 \)
\( \Rightarrow\left(k^{2}+4\right)=0 \quad \) und \( \left(k^{2}+4\right)=0 \)
$$ k_{2 / 3}=\pm 2_{i} \quad k_{4 / 5}=\pm 2_{i} $$
\( y_{2}=c_{2} \cos (2 x) \)
\( y_{3}=c_{3} x \cdot \cos (2 x) \)
\( y_{4}=c_{4} \sin (2 x) \)
\( y_{5}=c_{5} x \cdot \sin(2 x)) \)
\( \Rightarrow y_{n}=c_{1} e x+c_{2} \cos (2 x)+ c_{3} x \cos (2 x) + c_{4} \sin (2 x)+c_{5} x \cdot \sin (2 x) \)
\( \Rightarrow y_{p}=A e^{3 x} \) usw.
\( \Rightarrow y=y_{n} + yp \)
