The following differential equation represents the motion of a body of mass $m$ and displacement $x$ from the mean position, that is attached to a spring of force constant $a$ and viscous damping coefficient $b$ :
$$\bbox[5px,border:1px solid black] {
 m\frac{d^2x}{dt^2}=-ax-b\frac{dx}{dt}}$$

On rearrangement of the above equation we get the following:
$$\bbox[5px,border:1px solid black] {
 \frac{d^2x}{dt^2}+\frac bm\cdot\frac{dx}{dt}+\frac amx=0
}$$
Defining $k\triangleq\frac{b}{2m}$ and $\omega^2=\frac am$, we have:
$$\bbox[5px,border:1px solid black] {
 \frac{d^2x}{dt^2}+2k\frac{dx}{dt}+\omega^2x=0
}$$

Hence, we solve it as:

            ![Image 1]](https://i.stack.imgur.com/EPOyR.jpg)

In the situation of the undamped oscillation (shown in the image below):

* What is the significance of the term $i(A-B)\sin(\omega't)$?
* And also is the line highlighted with green justified? Why or why not?

               ![Image 2](https://i.stack.imgur.com/yMwBj.jpg)