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5012(1)given IVP is\begin{aligned}& \frac{d^{2} y}{d t^{2}}+8 \frac{d y}{d t}+80 y=0 \\& y(0)=0, y^{\prime}(0)=9 \\\Rightarrow & y^{\prime \prime}(t)+8 y^{\prime}(t)+80 y(t)=0\end{aligned}Let L\{y(t)\}=y(s)\begin{array}{l}\Rightarrow L\left\{y^{\prime \prime}(x)+8 y^{\prime}(t)+80 y(x)\right\}=0 \\\Rightarrow s^{2} y(s)-s y^{\prime}(0)-y^{\prime}(0)+8 s y(s)-8 y(0)+80 y(s)=0 \\\Rightarrow s^{2} y(9)-s \times 0-g+8 s y(s)-8 \times 0+80 y(s)=0 \\\Rightarrow\left(s^{2}+8 s+80\right) y(s)=9 \\\Rightarrow y(s)=\frac{9}{S^{2}+8 s+80} \\\Rightarrow L\{y(t)\}=y(s)=\frac{9}{s^{2}+8 s+80} \text { Ans } \\\because y(5)=\frac{9}{5^{2}+88+80} \\=\frac{9}{5^{2}+2 \cdot 5 \cdot 4+16+80-16} \\\end{array}y(s)=\frac{s}{(s+4)^{2}+64}(2)Taking inverse Laplace on both sides weget\begin{aligned}y(t) & \left.=L \frac{9}{(s+4)^{2}+64}\right\} \\& =9 e^{-4 t} L^{-1}\left\{\frac{1}{s^{2}+b t}\right\} \\& =9 e^{-4 t} L^{-1}\left\{\frac{1}{s^{2}+8^{2}}\right\} \\& \left.=9 e^{-4 t} \cdot \frac{\sin 8 t}{8}\left\{\because 1-\frac{1}{s^{2}+a^{2}}\right\} \frac{\sin a t}{a}\right\} \\& =\frac{9}{8} e^{-4 t} \cdot \sin (8 t) \\\therefore y(t) & =\frac{9}{8} e^{-4 t} \cdot \sin (8 t)\end{aligned} ...