Convert Second-order ODE to First-order Linear System

Convert a second-order linear ODE to a first-order linear system of ODEs and rewrite this system as a matrix equation.

[fahmyeby]

You did what my Professor did for a whole semester in 4 minutes. God bless you

[Darieee]

Thanks !

Very annoying that everyone uses a completely different notation when it comes to diff equations .. but .. it is what it is

Awesome video too !

[ivanjaramillo]

Really great stuff! Clarified any questions I had and showed me different notations in case people didn't write the way I write. Thank you.

[blacklite]

Thank you ! I spent 2 hours watching my lecture and my professor couldn't explain properly how to convert them.

[waynelast]

interesting but what happens if y-prime equals zero?

[gsairforce]

How does it work when it's not a homogenous differential equation ? I have to convert: y''(t) - 3y'(t) + y(t) = cos(t) but I don't know what to do with the cos(t).

[eldi]

What are the requirements for the differential equation that one wishes to convert?
Must it be homogeneous?
Can it have non-constant coefficients?
Must it be linear?
Can the same be done for partial differential equations?

[Scott]

Thanks for the video. However, what do you do if you have just one y'' in the equation? For example: y'' = -f(L-x)?

[victorpaesplinio]

Can this method be generalized to convert a system of two 2nd order ODE to a system of four 1st order ODE?

[sidtv]

Can you do this with nonlinear ODE's where f(t) is a constant?

[pmcate]

Ok but why relabel y as u? Seems a bit unecessary

[pt-ring]

How did 5 become positive in the A matrix from starting negative?

[tentonk]

Would [1 0 ; 5 -6] also be valid? I rearranged it where v was first instead of u.