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Use the method of undetermined coefficients to find the general solution
of the differential equation (Do not find the coefficients)
$$y^{\prime \prime}+8 y^{\prime}+16 y=2 \cos h(4 x)$$

$$\left(D^{2}+8 D+16\right) y=2 \cdot \frac{e^{4 x}+e^{-4 x}}{2}=e^{4 x}+e^{-4 x}$$

$$\rightarrow\left(D^{2}+8 D+16\right) y=e^{4 x}+e^{-4 x}$$

(1) $$y \: c : f(m)=0 \rightarrow D^{2}+8 D+16=0$$

Replace $$D$$ by $$m \quad m^{2}+8 m+16=0$$

$$(m+4)(m+4)=0 \rightarrow m$$ -values : $$-4,-4$$

$$y {c}=\left(c_{1}+c_{2} x\right) e^{-4 x}$$

(2) $$y {p}=A e^{4 x}+B x^{2} e^{-4 x}$$

$$y_{G \cdot s}=y {c}+y p$$

$$=\left(c_{1}+c_{2} x\right) e^{-4 x}+A e^{4 x}+B x^{2} e^{-4 x}$$

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