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Solutron:- Given Differuitial Equations^{\prime \prime}+b s^{l}+38=0Naw the charachrstic equation far the abaue differutial equatian ism^{2}+b u+3=0Far the abaue quadrate equation\begin{aligned}b^{2}-4 c & =b^{2}-4 \cdot 3 \\& =b^{2}-12\end{aligned}a) far auer dampeal syolen\begin{array}{c}b^{2}-4 c>0 \\b^{2}-12>0 \\(b-\sqrt{12})(b+\sqrt{12})>0 \\b \alpha-\sqrt{12} \text { ar } b>\sqrt{12} \\b<-2 \sqrt{3} \text { or } b>2 \sqrt{3}\end{array}i.ie far aur clamped syptem b \in(-\infty,-2 \sqrt{3}) \cup(2 \sqrt{3}, \infty)b) For under damped syoums\begin{array}{l}b^{2}-4 c<0 \\b^{2}-12<0 \\(b-\sqrt{12})(b+\sqrt{12})<0 \\-\sqrt{12} \alpha b<\sqrt{12} \text { or }-2 \sqrt{3}<b<2 \sqrt{3}\end{array}lie far under damped Syolum b \in(-2 \sqrt{3}, 2 \sqrt{3})c) For Gritically damped syotin:\begin{array}{l}b^{2}-4 c=0 \\b^{2}-12=0 \\b^{2}=12 \quad \Rightarrow b=\sqrt{12} \\b=2 \sqrt{3}\end{array}ie b=2 \sqrt{3} far Gutically damped syotrm ...