Solution to a 2nd order, linear homogeneous ODE with repeated roots

I discuss and solve a 2nd order ordinary differential equation that is linear, homogeneous and has constant coefficients.
In particular, I solve
$$y'' - 4y' + 4y = 0.$$
The solution method involves reducing the analysis to the roots of of a quadratic (the characteristic equation). Such an example is seen in 1st and 2nd year university mathematics.

[Alibey]

Chris,
You are doing an excellent job with your videos. I've been re-learning Math thru your ODE, Laplace and other videos. Compared to other videos in youtube, it is nice to see the person that teaches rather than only his/her voice.
Thank you from Lafayette, Indiana
Ali.

[blackphoenix]

i was in ur maths 1B back in 2007 - so when i had to relearn ODE i thought i;d google to see if u have ur lecture notes. im SO glad u have these vids up!! major help! thanks =)

[JamesEnglund]

The greatest maths teacher on earth. I studied physics at UNSW... where were you all those years! :) Thanks for these videos Chris

[DrChrisTisdell]

Yes, I tend to feel more comfortable with a "live" audience! Later this session i will be experimenting with video recording of full lectures. Watch this space!

[DrChrisTisdell]

You're right, Iruka. The sound and lighting could be improved. Some of the more recently uploaded vids have better sound and lighting as I've been experimenting with simple post-production.

Thanks for posting!

[JeMiodhih]

very very nice video!! thanks a lot, great help! would have been better if you had also showed the particular solution for the example also!

[Ensign_Cthulhu]

You could just as easily write the general form for repeated roots as y(x) = (e^lambda.x) [A+Bx}, couldn't you? This would differentiate it from the distinct-roots solution and ensure that the x term attached to the B coefficient isn't forgotten.

[DrChrisTisdell]

Haha - I started this project in 2008, so your studies may pre-date my YT adventures.
Very humbling comment! :-)

[jdrabble]

Is there a better explanation as to why we add an x to one of the exponentials if we have a repeated root?

[de]

you looks nervous in some videos nevertheless very helpful videos for those hard 2019 tute problems.

perhaps your in class lecture videos could be helpful too.