Question (1 point) For the differential equation $" + bs' + 7s = 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< 0o, 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

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Transcribed Image Text: (1 point) For the differential equation $" + bs' + 7s = 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

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Solution\Rightarrow The characterstic equation givens^{\prime \prime}+b s^{\prime}+7 s=0\Rightarrow the above equation rewrit quadratic form.t^{2}+b t+7=0 \quad \text { quabric equation }here a=1, b=b, c=7 \quad\left(" a x^{2}+b x+c=0\right)(a) For over-damped, b^{2}-4 a c>0\begin{aligned}\Rightarrow & b^{2}-28>0 \\\Rightarrow & (b-\sqrt{28})(b+\sqrt{28})>0 \\& b<-\sqrt{28} \text { or } b>\sqrt{28} \\& (b<-2 \sqrt{7}) \text { or }(b>2 \sqrt{7}) \rightarrow \text { over-damped }\end{aligned}over-damped(b) For under-damped, b^{2}-4 a c<0\begin{array}{l}b^{2}-28<0 \\(b-\sqrt{28})(b+\sqrt{28})<0 \\b<\sqrt{28} \text { or }-\sqrt{28}>b \\-2 \sqrt{7}<b<2 \sqrt{7} \text { under damped }\end{array}(c) for crifically damped b^{2}-4 a c=0\begin{array}{l}\left.b^{2}-d^{28}\right)=0 \quad \Rightarrow b=\sqrt{28} \\b=2 \sqrt{7} \quad \text { critical-damped }\end{array} ...