Answer:
The answer is
"
".
Explanation:
Taking into consideration a volume weight = 16 pounds originally extends a springs
feet but is extracted to resting at 2 feet beneath balance position.
The mass value is =
![W=mg\\m=\frac{w}{g}\\m=\frac{16}{32}\\m= \frac{1}{2} slug\\](https://tex.z-dn.net/?f=W%3Dmg%5C%5Cm%3D%5Cfrac%7Bw%7D%7Bg%7D%5C%5Cm%3D%5Cfrac%7B16%7D%7B32%7D%5C%5Cm%3D%20%5Cfrac%7B1%7D%7B2%7D%20slug%5C%5C)
The source of the hooks law is stable,
![16= \frac{8}{3} k \\\\8k=16 \times 3 \\\\k=16\times \frac{3}{8} \\\\k=6 \frac{lb}{ft}\\\\](https://tex.z-dn.net/?f=16%3D%20%5Cfrac%7B8%7D%7B3%7D%20k%20%5C%5C%5C%5C8k%3D16%20%5Ctimes%203%20%5C%5C%5C%5Ck%3D16%5Ctimes%20%5Cfrac%7B3%7D%7B8%7D%20%5C%5C%5C%5Ck%3D6%20%5Cfrac%7Blb%7D%7Bft%7D%5C%5C%5C%5C)
Number
times the immediate speed, i.e .. Damping force
![\frac{1}{2} \frac{d^2 x}{dt^2} = -6x-\frac{1}{2}\frac{dx}{dt}+10 \cos 3t \\\\\frac{1}{2} \frac{d^2 x}{dt^2}+ \frac{1}{2}\frac{dx}{dt}+6x =10 \cos 3t \\ \\\frac{d^2 x}{dt^2} +\frac{dx}{dt}+12x=20\cos 3t \\\\](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7B2%7D%20%5Cfrac%7Bd%5E2%20x%7D%7Bdt%5E2%7D%20%3D%20-6x-%5Cfrac%7B1%7D%7B2%7D%5Cfrac%7Bdx%7D%7Bdt%7D%2B10%20%5Ccos%203t%20%5C%5C%5C%5C%5Cfrac%7B1%7D%7B2%7D%20%20%5Cfrac%7Bd%5E2%20x%7D%7Bdt%5E2%7D%2B%20%5Cfrac%7B1%7D%7B2%7D%5Cfrac%7Bdx%7D%7Bdt%7D%2B6x%20%3D10%20%5Ccos%203t%20%5C%5C%20%5C%5C%5Cfrac%7Bd%5E2%20x%7D%7Bdt%5E2%7D%20%2B%5Cfrac%7Bdx%7D%7Bdt%7D%2B12x%3D20%5Ccos%203t%20%5C%5C%5C%5C)
The m^2+m+12=0 and m is an auxiliary equation,
![m=\frac{-1 \pm \sqrt{1-4(12)}}{2}\\\\m=\frac{-1 \pm \sqrt{47i}}{2}\\\\\ m1= \frac{-1 + \sqrt{47i}}{2} \ \ \ \ or\ \ \ \ \ m2 =\frac{-1 - \sqrt{47i}}{2}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-1%20%5Cpm%20%5Csqrt%7B1-4%2812%29%7D%7D%7B2%7D%5C%5C%5C%5Cm%3D%5Cfrac%7B-1%20%5Cpm%20%5Csqrt%7B47i%7D%7D%7B2%7D%5C%5C%5C%5C%5C%20m1%3D%20%5Cfrac%7B-1%20%2B%20%5Csqrt%7B47i%7D%7D%7B2%7D%20%5C%20%5C%20%5C%20%5C%20or%5C%20%5C%20%5C%20%5C%20%5C%20%20m2%20%3D%5Cfrac%7B-1%20-%20%5Csqrt%7B47i%7D%7D%7B2%7D)
Therefore, additional feature
![x_c (t) = e^{\frac{-t}{2}}[C_1 \cos \frac{\sqrt{47}}{2}t+ C_2 \sin \frac{\sqrt{47}}{2}t]](https://tex.z-dn.net/?f=x_c%20%28t%29%20%3D%20e%5E%7B%5Cfrac%7B-t%7D%7B2%7D%7D%5BC_1%20%5Ccos%20%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%2B%20C_2%20%5Csin%20%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%5D)
Use the form of uncertain coefficients to find a particular solution.
Assume that solution equation,
![x_p = Acos(3t)+B sin(3t) \\x_p'= -3A sin (3t) + 3B cos (3t)\\x_p}^{n= -9 Acos(3t) -9B sin (3t)\\](https://tex.z-dn.net/?f=x_p%20%3D%20Acos%283t%29%2BB%20sin%283t%29%20%5C%5Cx_p%27%3D%20-3A%20sin%20%283t%29%20%2B%203B%20cos%20%283t%29%5C%5Cx_p%7D%5E%7Bn%3D%20-9%20Acos%283t%29%20-9B%20sin%20%283t%29%5C%5C)
These values are replaced by equation ( 1):
![\frac{d^2x}{dt}+\frac{dx}{dt}+ 12x=20 \cos(3t) -9 Acos(3t) -9B sin (3t) -3Asin(3t)+3B cos (3t) + 12A cos (3t) + 12B sin (3t)\\\\3Acos 3t + 3B sin 3t - 3Asin 3t + 3B cos 3t= 20cos(3t)\\(3A+3B)cos3t -(3A-3B)sin3t = 20 cos (3t)\\](https://tex.z-dn.net/?f=%5Cfrac%7Bd%5E2x%7D%7Bdt%7D%2B%5Cfrac%7Bdx%7D%7Bdt%7D%2B%2012x%3D20%20%5Ccos%283t%29%20-9%20Acos%283t%29%20-9B%20sin%20%283t%29%20-3Asin%283t%29%2B3B%20cos%20%283t%29%20%2B%2012A%20cos%20%283t%29%20%2B%2012B%20sin%20%283t%29%5C%5C%5C%5C3Acos%203t%20%2B%203B%20sin%203t%20-%203Asin%203t%20%2B%203B%20cos%203t%3D%2020cos%283t%29%5C%5C%283A%2B3B%29cos3t%20-%283A-3B%29sin3t%20%3D%2020%20cos%20%283t%29%5C%5C)
Going to compare cos3 t and sin 3 t coefficients from both sides,
The cost3 t is 3A + 3B= 20 coefficients
The sin 3 t is 3B -3A = 0 coefficient
The two equations solved:
![3A+3B = 20 \\\frac{3B -3A=0}{}\\6B=20\\B= \frac{20}{6}\\B=\frac{10}{3}\\](https://tex.z-dn.net/?f=3A%2B3B%20%3D%2020%20%5C%5C%5Cfrac%7B3B%20-3A%3D0%7D%7B%7D%5C%5C6B%3D20%5C%5CB%3D%20%5Cfrac%7B20%7D%7B6%7D%5C%5CB%3D%5Cfrac%7B10%7D%7B3%7D%5C%5C)
Replace the very first equation with the meaning,
![3B -3A=O\\3(\frac{10}{3})-3A =0\\A= \frac{10}{3}\\](https://tex.z-dn.net/?f=3B%20-3A%3DO%5C%5C3%28%5Cfrac%7B10%7D%7B3%7D%29-3A%20%3D0%5C%5CA%3D%20%5Cfrac%7B10%7D%7B3%7D%5C%5C)
equation is
![x_p\\\\\frac{10}{3} cos (3 t) + \frac{10}{3} sin (3t)](https://tex.z-dn.net/?f=x_p%5C%5C%5C%5C%5Cfrac%7B10%7D%7B3%7D%20cos%20%283%20t%29%20%2B%20%5Cfrac%7B10%7D%7B3%7D%20sin%20%283t%29)
The ultimate plan for both the equation is therefore
![x(t)= e^\frac{-t}{2} (c_1 cos \frac{\sqrt{47}}{2}t)+c_2\sin\frac{\sqrt{47}}{2}t)+\frac{10}{3}\cos (3t)+\frac{10}{3}\sin (3t)](https://tex.z-dn.net/?f=x%28t%29%3D%20e%5E%5Cfrac%7B-t%7D%7B2%7D%20%28c_1%20cos%20%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%29%2Bc_2%5Csin%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%29%2B%5Cfrac%7B10%7D%7B3%7D%5Ccos%20%283t%29%2B%5Cfrac%7B10%7D%7B3%7D%5Csin%20%283t%29)
Initially, the volume of rest x(0)=2 and x'(0) is extracted by rest i.e.
Throughout the general solution, replace initial state x(0) = 2,
Replace x'(0)=0 with a general solution in the initial condition,
![x(t)= e^\frac{-t}{2} [(c_1 cos \frac{\sqrt{47}}{2}t)+c_2\sin\frac{\sqrt{47}}{2}t)+\frac{10}{3}\cos (3t)+\frac{10}{3}\sin (3t)]\\\\](https://tex.z-dn.net/?f=x%28t%29%3D%20e%5E%5Cfrac%7B-t%7D%7B2%7D%20%5B%28c_1%20cos%20%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%29%2Bc_2%5Csin%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%29%2B%5Cfrac%7B10%7D%7B3%7D%5Ccos%20%283t%29%2B%5Cfrac%7B10%7D%7B3%7D%5Csin%20%283t%29%5D%5C%5C%5C%5C)
![x(t)= e^\frac{-t}{2} [(-\frac{\sqrt{47}}{2}c_1\sin\frac{\sqrt{47}}{2}t)+ (\frac{\sqrt{47}}{2}c_2\cos\frac{\sqrt{47}}{2}t)+c_2\cos\frac{\sqrt{47}}{2}t) +c_1\cos\frac{\sqrt{47}}{2}t +c_2\sin\frac{\sqrt{47}}{2}t + \frac{-1}{2}e^{\frac{-t}{2}} -10 sin(3t)+10 cos(3t) \\\\](https://tex.z-dn.net/?f=x%28t%29%3D%20e%5E%5Cfrac%7B-t%7D%7B2%7D%20%5B%28-%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dc_1%5Csin%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%29%2B%20%28%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dc_2%5Ccos%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%29%2Bc_2%5Ccos%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%29%20%20%2Bc_1%5Ccos%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%20%2Bc_2%5Csin%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%20%2B%20%5Cfrac%7B-1%7D%7B2%7De%5E%7B%5Cfrac%7B-t%7D%7B2%7D%7D%20-10%20sin%283t%29%2B10%20cos%283t%29%20%5C%5C%5C%5C)
![c_2=\frac{-64\sqrt{47}}{141}](https://tex.z-dn.net/?f=c_2%3D%5Cfrac%7B-64%5Csqrt%7B47%7D%7D%7B141%7D)
![x(t)= e^\frac{-t}{2}((\frac{-4}{3})\cos\frac{\sqrt{47}}{2}t- \frac{-64\sqrt{47}}{141} \sin\frac{\sqrt{47}}{2}t)+\frac{10}{3}(\cos(3t)+ \sin (3t))](https://tex.z-dn.net/?f=x%28t%29%3D%20e%5E%5Cfrac%7B-t%7D%7B2%7D%28%28%5Cfrac%7B-4%7D%7B3%7D%29%5Ccos%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt-%20%5Cfrac%7B-64%5Csqrt%7B47%7D%7D%7B141%7D%20%5Csin%5Cfrac%7B%5Csqrt%7B47%7D%7D%7B2%7Dt%29%2B%5Cfrac%7B10%7D%7B3%7D%28%5Ccos%283t%29%2B%20%5Csin%20%283t%29%29)