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• Notes

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) $y {p}=A e^{4 x}+B x^{2} e^{-4 x}$

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

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