[Python] How to animate or visualise the ±cosine(wt) solutions to d²x/dt²= -w²x

I'm currently studying simple harmonic motion and trying to understand the general solution to the differential equation:

d²x/dt² = -ω²x

The solution is:

x(t) = ±A·cos(ωt)

This ± sign has led to a conceptual doubt. Mathematically, squaring and taking the square root introduces both positive and negative branches, but physically we only see one path for the oscillator.

I want to graphically or interactively explore what it means to include both +cos(ωt) and -cos(ωt) as solutions. Specifically, I want to:

1. 
   Visualize both branches (positive and negative cosine solutions).
   
   
2. 
   Animate the point moving on both curves (if that makes sense).
   
   
3. 
   Optionally: show how taking ± when integrating square root can lead to bifurcated paths, but we choose one physically.
   

Here what I’ve already tried in Python using matplotlib (and it works for static plot):

But now I want to go beyond static plots:

Can this be animated? (in Python, JavaScript, or even Manim?)

How can I visually show that the ± sign is mathematical, but we choose one path physically based on initial conditions (e.g., x(0) = A)?

What tool (e.g., Desmos, Python animation, p5.js, Manim) would you recommend? Also apologise if this is not the correct site to ask it

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