Question (1 point) The following differential equations represent oscillating springs. = = = = - (i) 9s" + s = 0, s(0) = 6, s'(0) = 0. (ii) s" + 36s = 0, s(0) = 2, s'(0) = 0. (iii) s" + 9s = 0, $(0) = 3, s'(0) = 0. (iv) 36s" + s = 0, $(0) = 12, s'(0) = 0. - - Which differential equation represents: (a) The spring oscillating most quickly (with the shortest period)? ? (b) The spring oscillating with the largest amplitude? ? (c) The spring oscillating most slowly (with the longest period)? ? (a) The spring oscillating with the largest maximum velocity?| ?

\begin{array}{l}s(0)=c_{1} \cos 0+c_{2} \sin 0=c_{1}=12 \\s^{\prime}(t)=\frac{-c_{1}}{6} \sin \left(\frac{t}{6}\right)+\frac{c_{2}}{6} \cos \left(\frac{t}{6}\right) \\s^{\prime}(0)=\frac{c_{2}}{6}=0 \Rightarrow c_{2}=0\end{array}Hence, s(t)=12 \cos \left(\frac{t}{6}\right)s(t)=A \cos (\omega t)=12 \cos \left(\frac{t}{6}\right)\Rightarrow Amplitude, A=12, \omega=\frac{1}{6}\begin{array}{l}T=\frac{2 \pi}{w}=\frac{2 \pi}{(1 / 6)}=12 \pi \\V_{\text {max }}=A w=12 \times \frac{1}{6}=2 .\end{array}a) The spring oscillating most quickly (with the shortest perid) is spring no. (ii) with time period \pi / 3=Tb) The spring oscillating with the largest amplitude is spring no. (iv) with amplitude A=12c) The spring oscillating most slowy (with the longest peri(d) is spring no.(iV) with T=12 \pid) The spring oscillating with the largest maximum velocity is sping no. (ii) with v_{\text {max }}=12u200bu200bu200bu200bu200bplease do upvote thankyou!!(i)\begin{array}{c}9 s^{\prime \prime}+s=0, s(0)=6, s^{\prime}(0)=0 \\s^{\prime \prime}=\frac{s}{9}=0 \\\Rightarrow \frac{d^{2} s}{d t^{2}}+\frac{s}{9}=0 \quad \text { where let } \frac{d}{d t}=D \\\Rightarrow\left[D^{2}+\frac{1}{9}\right] s=0 \quad \Rightarrow \quad D^{2}+\frac{1}{9}=0\end{array}So, D= \pm \sqrt{-\frac{1}{9}}= \pm \frac{1}{3} i= \pm \frac{i}{3}D=\frac{ \pm i}{3}=\alpha \pm i \beta \Rightarrow \alpha=0 \text { and } \beta=\frac{1}{3}so, S(t)=\left(c_{1} \cos \beta t+c_{2} \sin \beta t\right) e^{\alpha t}\Rightarrow s(t)=1_{1} \cos \left(\frac{t}{3}\right)+c_{2} \sin \left(\frac{t}{3}\right) \text { where } e^{\circ}=1compare with s(t)=c_{1} cosut +c_{2} sinut\begin{array}{l} \rightarrow \omega=\frac{1}{3} \\S(0)=6 \Rightarrow c_{1} \cos 0+c_{2} \sin 0=6 \\\Rightarrow c_{1}=6\end{array}\begin{aligned}s^{\prime}(t) & =-\frac{c_{1}}{3} \sin \left(\frac{t}{3}\right)+\frac{c_{2}}{3}\left(\frac{t}{3}\right) \\s^{\prime}(0) & =-\frac{c_{1}}{3} \sin (0)+\frac{c_{2}}{3} \cos (0) \\\Rightarrow s^{\prime}(0) & =\frac{c_{2}}{3}=0 \Rightarrow c_{2}=0\end{aligned}Hence, s(t)=6 \cos \left(\frac{t}{3}\right)\Rightarrow s(t)=A \cos (\omega t)=6 \cos \left(\frac{t}{3}\right)So, Amplitude, A=6, \omega=\frac{1}{3} \Rightarrow Time period (t)=\frac{2 \pi}{\omega}\Rightarrow T=\frac{2 \pi}{(1 / 3)}=6 \pi \quad \varepsilon maximum veloity,V_{\max }=A C==6 \times \frac{1}{3}=2(ii)\begin{array}{l}s^{\prime \prime}+36 s=0, s(0)=2 \quad s^{\prime}(0)=0 \\\frac{d^{2} s}{d t^{2}}+36 s=0 \Rightarrow\left(D^{2}+36\right) s=0 \\\text { Henc, } \quad D= \pm \sqrt{-36}= \pm 6 i\end{array}Henc, D= \pm \sqrt{-36}= \pm 6 i\begin{array}{l}s(t)=c_{1} \cos 6 t+c_{2} \sin 6 t \\s(0)=c_{1}=2 \text { and } s^{\prime}(t)=-6 c_{1} \sin 6 t+6 c_{2} \cos 6 t \\s^{\prime}(0)=6 c_{2}=0 \Rightarrow c_{2}=0\end{array}Hence, s(t)=2 \cos 6 t=A \cos (\omega t)\Rightarrow Amplitude, A=2, \omega=6\begin{array}{l}p=\frac{2 \pi}{\omega}=\frac{2 \pi}{6}=\pi / 3 \\V_{\max }=A W=2 \times 6=12\end{array}(iii)\begin{array}{l}s^{\prime \prime}+9 s=0, s(0)=3, \quad s^{\prime}(0)=0 \\\left(D^{2}+9\right) s=0 \Rightarrow \frac{d^{2} s}{d t^{2}}+9 s=0 \\\Rightarrow D^{2}+9=0 \text { which gives } D= \pm \sqrt{-3}= \pm 3 i \\s(t)=c_{1} \cos 3 t+c_{2} \sin 3 t \Rightarrow \quad \text { and } s^{\prime}(t)=-3 c_{1} \sin 3 t+3 c_{2} \cos 3 t \\s(0)=C_{1}=3 \quad s^{\prime}(0)=0 \Rightarrow c_{2}=0\end{array}Henc, s(t)=3 \cos 3 t=A \cos \omega tSo, A=3, w=3, \quad T=\frac{2 \pi}{3}, V_{\max }=3 \times 3=9(iv)\begin{array}{l}36 s^{\prime \prime}+s=0, \quad s(0)=12, \quad s^{\prime}(0)=0 \\s^{\prime \prime}+\frac{s}{36}=\frac{d^{2} s}{d t^{2}}+\frac{s}{36}=0 \\\Rightarrow\left(D^{2}+\frac{1}{36}\right) 5=0 \\s 0, D= \pm \sqrt{-\frac{1}{36}}= \pm \frac{1}{6} i=\alpha \pm i \beta \Rightarrow \alpha=0, \beta=\frac{1}{6} \\s(t)=c_{1} \cos \frac{t}{6}+c_{2} \sin \frac{t}{6}\end{array} ...