Question (1 point) For the differential equation s" + bs' + 65 = 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 < o, 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
Solved 1 Answer
See More Answers for FREE
Enhance your learning with StudyX
Receive support from our dedicated community users and experts
See up to 20 answers per week for free
Experience reliable customer service
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 ...