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Solution 1 \rightarrow Guven Second order equations^{\prime \prime}+6 s+6 s=0roots of the characterustic eq " of the ODE given above are\begin{aligned}S_{1,2} & =\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} \\\Rightarrow S_{1,2} & =\frac{-b \pm \sqrt{b^{2}-4 \times 1 \times 6}}{2}\end{aligned}for our damped case, roots nust be real and destinict. Hence \quad b^{2}-24>0\begin{aligned}& (b-2 \sqrt{6})(b+2 \sqrt{6})>0 \\\Rightarrow & b>2 \sqrt{6} \& b<-2 \sqrt{6} \\\Rightarrow & b \in(-\infty,-2 \sqrt{6}),(2 \sqrt{6}, \infty)\end{aligned}For crifically demped case s_{1}=s_{2}=\frac{-b}{2} in the case b^{2}-24=0 \Rightarrow b=-2 \sqrt{6}, 2 \sqrt{6} Hence b \in[-2 \sqrt{6},-2 \sqrt{6}],[2 \sqrt{6}, 2 \sqrt{6}]- ArswerPor under - damped system b^{2}-24<0\Rightarrow(b-2 \sqrt{6})(b+2 \sqrt{6})<0 \Rightarrow-2 \sqrt{6}<b<2 \sqrt{6}Hence for under-damped systemb \in(-2 \sqrt{6}, 2 \sqrt{6})part(b) answer ...