Drawing Direction Fields for Higher Order ODEs

[hagopbul]

TL;DR

Direction field of higher order differential equation

Hello :

Trying to find references on drawing direction fields of higher order differential equation by hand as 1st step then by computer , do you know any reference I can read ( PDF , books ,...) , and hope it is not only some short notes

Best regards
HB

 

[DaveE]

Have you tried google search yet?

 

[hagopbul]

Thank you but this is only first order ode I was looking for higher order ode , and that video clip I already watched

 

[fresh_42]

Maybe these examples can help you
https://web.sha1.bfh.science/Labs/F2/ODE/VectorFields.pdf

 

[DaveE]

This maybe?
http://howellkb.uah.edu/DEtext/Additional_Chapters/Part6/Trajectories.pdf

There's lots of this stuff in the state space world. Sometimes the names are different, like phase portraits, trajectories, slope fields, etc.

 

[hagopbul]

thanks but all the files and the links that you provided only deal with 1st order ode , or computerize second order ode direction field , but what about how to draw direction field for second or higher differential equations by hand ? and for computerized solution what algorithm they are using as it is not clear in the code provided

 

[DaveE]

thanks but all the files and the links that you provided only deal with 1st order ode , or computerize second order ode direction field , but what about how to draw direction field for second or higher differential equations by hand ? and for computerized solution what algorithm they are using as it is not clear in the code provided

You may want to study a bit about the state space formulation of systems. In this treatment higher order systems are described as a system of first order DE's, so the "order" of the system is expressed in the number of coupled equations. For example, look at the section labeled "Higher Order Systems" in the link I provided.

I'm sure there are people that don't use this structure, but I'm not the person to ask about that. I only have experience with this, most common, formalism. Somehow, you must identify the state of your system to describe what it will do next. This will require a multi-dimensional basis for your field.

Also, there is a problem with drawing much in systems above order 2 or maybe 3. The theory is good in higher dimensions, but I don't know how to draw 4 dimensional plots. Nearly every example you will see is a 1st or 2nd order system.