Linearize a Differential Equation

[@saitohasegawa3294]

Thank you!! Your explanation made it so much simpler for me to grasp

[@AyaAya-fh2wx]

Many thanks for your work!!! Great video

[@bachircherafeddinebenhacin4125]

Thank you for the video, very well explained !!. Could you Please do an example of how to approximate the solution of a nonlinear Differential Equation using 2nd order polynomials (or using any given order of accuracy)?

[@apm]

apmonitor.com/do/index.php/Main/OrthogonalCollocation

[@dume85]

dx/dt=x would already be linear so there would be no need to do this correct?

[@apm]

Yes, it is already linear. Sometimes it is good to go through this process to get the model into "deviation variable" form.

[@AJ-et3vf]

Great video sir! Thank you!

[@meghnagoyal224]

Hi, thanks for the video. I did not understand why f(x) is written as f(xo) + df/dx [the first step basically, where this value is plugged into the first equation dx/dt = f(x).

[@apm]

It is the first term of a Taylor series approximation of the right hand side of the equation. Here is additional info: apmonitor.com/pdc/index.php/Main/ModelLinearization

[@mariahanif9664]

how we linearized non linear partial differential equation

[@apm]

Here is a general method that can also be applied to PDEs: apmonitor.com/pdc/index.php/Main/ModelLinearization

[@Meloman0001]

Well explained. Thank you.

[@gilpeleg2316]

Thank you very much, very well explained

[@samardeepsinghsarna8091]

What exactly is z storing, why is it a 2x2 matrix? How are dimensions of x1 x2 related to z?

[@apm]

The variable z at 18:10 is a vector (1 dimension length 2, not 2x2). The function needs a vector of state values of the model as a list. There are additional tutorials on integrating systems of differential equations at apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations Please see Problems 3 and 4.

[@fatmas1996]

Thanks so much for your explanation ..

[@abidalrk4432]

You forgot to mention that the linearized equation is in the neighborhood of x0

[@apm]

Thanks for the additional comment. You are correct.

[@samardeepsinghsarna8091]

Dimension of x1 and x2 is equal to z or equal to t?

[@apm]

At 18:25 all of the x1, x2, and t are scalar values. After the solution is returned, you can parse out the individual solutions as column vectors. They will have the same length as the time points that you requested from ODEINT.

[@samardeepsinghsarna8091]

@@apm Thanks. and what about z. How does it become 2x2 dimensional and what are the 2 rows representing?

[@rrc]

@@samardeepsinghsarna8091, here is help with ODEINT apmonitor.com/pdc/index.php/Main/SolveDifferentialEquations please see example 4 for a similar looping function to integrate the differential equations

[@naturalselection4645]

Awesome 😎

[@cool20guitar]

Thanks a lot!!!

[@jarrydnii9993]

Awesome man! Thanks

[@kanyetaernest9540]

Thanks loads

[@johnkurtanidze3284]

This is a multivariable equation example you showed, not a diff eq smh

[@apm]

You are correct that it is multivariate. The differential is on the left side of the equation. See apmonitor.com/pdc/index.php/Main/ModelLinearization for more info.

[@johnkurtanidze3284]

@@apm Yeah I've been on that website before. It's just as wrong as your video.