Three Solutions for a Simple Harmonic Oscillator (with initial conditions)

Consider a simple harmonic oscillator in 1D. Here are three solutions that satisfy the differential equation.
Here is my playlist with all the oscillator videos
• Harmonic Oscillators
00:00 - Introduction
01:37 - Example Motion in Python
03:08 - Solution 1: Sine and Cosine
10:31 - Checking Solution 1
13:22 - Solution 2: Cosine with phase shift
18:49 - Checking Solution 2
20:47 - Solution 3: Exponentials

[@olivioarmandocordeirojunio8280]

Thanks for sharing. Wonderful explanation.

[@madmister]

I dont understand what does a and b physiically represent if i m given an pendulum or spring mass system help me or my brain will fry

[@AbdulHameed-ge4sc]

Nice explained sir , as you have proved the two different solutions equivalent, what about the third one , is that also equivalent to the other two?

[@DotPhysics]

Yes, they are all solutions to the same differential equation. You can show that the e^iwt version is the same as sin and cosine version using the Euler equation. Maybe I'll make an update.

[@nigarabdulkalam2130]

Sir can u explain Numerical simple harmonic oscillator using eigenvalue and eigenvector

[@designX404]

amazing explanation , question: in 2nd equation why did you select A(cos wt + phi) why not A(sin wt + phi) and both equations will give negative sign in X double dot?? , why use COS instead of SIN

[@tomk4756]

to the best of my knowledge I think you can use either, but it will change the phi value, as the sin and cosine standard functions we are used to are the same if shifted by pi radians.