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A $16-lb$ weight is attached to a lower end of the coil spring suspended to the ceiling. The weight comes to rest in its equilibrium position thereby stretching the spring about $0.4 ft$ At the beginning of $t=0$ an external force is driving the system to force it to oscillate and is given by the $F(t)=40 \cos 16 t$ The medium offers resistance numerically equivalent to $4\frac{dx}{dt}$. $\frac{dx}{dt}$ is called the instantaneous velocity.

a)Seek the displacement of the weight as a function of time.

My work $$\frac{1}{2}\frac{d^2x}{dt^2}+4\frac{dx}{dt}+40x=40 \cos 16 t$$

$$ \frac{d^2x}{dt^2}+8\frac{dx}{dt}+80x=80 \cos 16 t $$

The complementary function, $$x_c=e^{-4t}[c_1\cos8t+c_1\sin8t]$$

Method of undetermined coefficients

$$y_p=A\cos 16 t+B\sin 16 t$$

$$A=-\frac{11}{37}$$

$$B=\frac{8}{37}$$

$$c_1=\frac{11}{37}$$

$$c_2=-\frac{21}{74}$$

The general solution is then,

$$x=e^{-4t}[\frac{11}{37}\cos 8 t-\frac{21}{74}\sin 8 t]-\frac{11}{37}\cos 16 t+ \frac{8}{37}\sin 16 t$$

b)Graph the transient state and that steady state separately. After that, find the real displacement graph.

I am not sure how to graph them

Transient state(it will get smaller and smaller then diminish) $$e^{-4t}[\frac{11}{37}\cos 8 t-\frac{21}{74}\sin 8 t]$$

Steady sate(the term will continue to oscillate at certain frequency) $$-\frac{11}{37}\cos 16 t+ \frac{8}{37}\sin 16 t$$

I can graph them separately but how should I combine them. ?Is there any trick to combine the graph

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To combine them, add the two parts of the function $x(t)$.

enter image description here

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