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$$y^{\prime \prime}+6 y^{\prime}+8 y=3 e^{-2 x}+2 x$$

(1) Find $$y_{c} :$$

$$y^{\prime \prime}+6 y^{\prime}+8 y=0 \quad f(m)=0$$

$$m^{2}+6 m+8=0 \rightarrow(m+4)(m+2)=0$$

$$m_{1}=-4 \quad, \quad m_{2}=-2 \rightarrow y_{c}=c_{1} e^{-4 x}+c_{2} e^{-2 x}$$

(2) Find yp: 

$$D^{2} \longrightarrow 2 x$$
$$D+2 \longrightarrow 3 e^{-2 x}$$


$$D^{2}(D+2)\left[D^{2}+6 D+8\right] y=D^{2}(D+2)\left[2 x+3 e^{-2 x}\right]$$

$$m^{2}(m+2)\left[m^{2}+6 m+8\right]=0$$


$$m_{1}=0, m_{2}=0, m_{3}=-2, m_{4}=-2, m_{5}=-4$$

$$y=c_{1}+c_{2} x+c_{3} e^{-2 x}+c_{4} x e^{-2 x}+c_5 e^{-4 x}$$

$$y_{c}=c_5 e^{-4 x}+c_3 e^{-2 x}$$

$$y p=c_{1}+c_2 x+c_4 x e^{-2 x}=A+B x+c x e^{-2 x}$$

$$ {y}^{\prime} p=B+C e^{-2{x}}+-C_{2} x e^{-2 x} $$

$$ y^{\prime \prime} p=-2 C e^{-2 x}-C_2 e^{-2 x}+C_{4} x e^{-2 x} $$

$$8 A+2 C e^{-2 x}+B(6+8 x)=3 e^{-2 x}+2 x$$

$$8 A+6 B+x(8 B)+e^{-2 x}(2 C)=3 e^{-2 x}+2 x$$

$$8 A+6 B=0$$

$$8 B=2 \rightarrow B=1 / 4$$

$$ \rightarrow \frac{8 A}{8}=-6\left(\frac{1}{4}\right)=-\frac{3}{2} / 8 \rightarrow A=\frac{-3}{16} $$

$$2 c=3 \rightarrow c=3 / 2$$

$$y p=\frac{-3}{16}+\frac{1}{4} x+\frac{3}{2} xe^{-2 x} $$

$$y_{g.s}=y_{c}+y p=c_{1}+e^{-4 x}+c_{2} e^{-2 x}+-\frac{3}{16}+\frac{1}{4} x+\frac{3}{2} x e^{-2 x} $$

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