Question (1 point) For the differential equation + bs' + 3s = 0, find the values of b that make the general solution overdamped, underdamped, or critically damped. (For each, give an interval or intervals for b for which the equation is as indicated. Thus if the the equation is overdamped for all b in the range -1<b< 1 and 3 < b < oo, enter (-1,1], [3,infinity); if it is overdamped only for b = 3, enter [3,3].) If the equation is overdamped, be If the equation is underdamped, be If the equation is critically damped, be

I have shown detailed solution below-Sol :-Given:-s^{\prime \prime}+b s^{\prime}+3 s=0solution of equation (1) will consist only complementary Function (C.F) part as particular Inte \operatorname{gral}(P \cdot I)=0solution for C.F\begin{aligned}&amp; \alpha^{2}+b \alpha+3=0 \\\therefore \alpha &amp; =\frac{-b \pm \sqrt{b^{2}-(4 \times 1 \times 3)}}{2 \times 1} \\&amp; =\frac{-b \pm \sqrt{b^{2}-12}}{2}\end{aligned}\left[\begin{array}{c}\text { where, } \alpha=\alpha_{1}, \alpha_{2} \\\text { are two roots }\end{array}\right]\rightarrow When the equation is overdamped, value of \left(b^{2}-12\right) must be greater than zero.i.e b^{2}>12or, b>2 \sqrt{3} \quad \therefore 2 \sqrt{3}<b \leq \infty.Therefore, b \in(2 \sqrt{3}, \infty)Ans (1)\rightarrow When the equation is underdamped, value of \left(b^{2}-12\right) must be less than zero.\text { i.e } b^{2}-12<0or, b<2 \sqrt{3}\begin{array}{l}=\frac{\text { co-efficient of } s}{2 \sqrt{(\text { corefticient of } s) \times\left(\text { co-efficient of } s^{\prime \prime}\right)}} \\=\frac{b}{2 \sqrt{1 \times 3}}=\frac{b}{2 \sqrt{3}}\end{array}Therefore, b should be greater than zero becausc if b=0 then S=0 i-e no damping.So, if the equation is underdampedb \in(0,2 \sqrt{3})Ans (2),\rightarrow For Critically damped\begin{array}{c}b^{2}-12=0 \\\text { or, } b=2 \sqrt{3}\end{array}Also, we can say for critically damped\left\{=1=\frac{b}{2 \sqrt{3}}\right.1 \text { or, } b=2 \sqrt{3}Therefore, if the equation is critically dampedb \in(2 \sqrt{3}, 2 \sqrt{3})Ans. (3) ...