Drawing Direction Fields for Higher Order ODEs

• I
• hagopbul
In summary, the person is trying to find references on how to draw direction fields for higher order differential equations by hand. They are looking for references that deal with 1st or computerized solution for second or higher order differential equations. There is a problem with drawing plots in higher dimensions.
hagopbul
TL;DR Summary
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

Have you tried google search yet?

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

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

hagopbul said:
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.

1. What are direction fields for higher order ODEs?

Direction fields for higher order ODEs are graphical representations of the behavior of a solution to a differential equation. They show the direction of the solution at different points on the graph and can help visualize the behavior of the solution over time.

2. How do you draw direction fields for higher order ODEs?

To draw direction fields for higher order ODEs, you first need to find the general solution to the differential equation. Then, choose a few values for the independent variable and use them to calculate the corresponding values for the dependent variable. Plot these points on a graph and draw short line segments at each point in the direction of the solution. Repeat this process for different values of the independent variable to create a more complete direction field.

3. Why are direction fields important in studying higher order ODEs?

Direction fields are important in studying higher order ODEs because they provide a visual representation of the behavior of the solution. This can help in understanding the characteristics of the solution and predicting its behavior over time. Direction fields can also help in identifying critical points and determining the stability of the solution.

4. Can direction fields be used to solve higher order ODEs?

No, direction fields cannot be used to directly solve higher order ODEs. They are used to visualize the behavior of the solution and provide insights into its characteristics. To find the exact solution to a higher order ODE, analytical or numerical methods must be used.

5. Are there any limitations to using direction fields for higher order ODEs?

Yes, there are some limitations to using direction fields for higher order ODEs. They are only applicable to autonomous differential equations, meaning the equation does not explicitly depend on the independent variable. Direction fields also cannot accurately represent solutions that are not continuous or have discontinuities. Additionally, direction fields may not show the complete behavior of the solution, especially for complex higher order ODEs.

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