How do I rewrite this system of coupled 2nd order ODEs into 1st order linear ODEs

[awaissssyed]

Theta and phi are functions of time (t). Need to rewrite these ODEs so that they can be solved.

Thank you for the help

 

[Deleted member 4993]

Theta and phi are functions of time (t). Need to rewrite these ODEs so that they can be solved.

Thank you for the helpView attachment 23449

Since you do not know where to start for these problems, please study the topic in the following web-site:

Differential Equations - Systems of Differential Equations

In this section we will look at some of the basics of systems of differential equations. We show how to convert a system of differential equations into matrix form. In addition, we show how to convert an nth order differential equation into a system of differential equations.

tutorial.math.lamar.edu

If you do not understand some steps - please comeback and ask specific questions.

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

READ BEFORE POSTING​

Please share your work/thoughts about this problem.

 

[topsquark]

This is similar to the spherical pendulum. So much so that I'm wondering if there may be some typos?

[math]\ddot{ \theta } = sin( \theta) ~ cos( \theta ) ~ \dot{ \phi } ^2 - \dfrac{g}{l} sin( \theta )[/math]
[math]\ddot{ \phi } = -2 ~ cot( \theta ) ~ \dot{ \theta } ~ \dot{ \phi }[/math]
This system is similar to the ordinary one dimensional pendulum (set [math]\phi (t) = 0[/math]), which has no exact solution. This in turn implies that the spherical pendulum has no exact solution, either. Did you want to solve the case for both [math]\theta (t) \text{ and } \phi (t)[/math] are small?

-Dan

 

[awaissssyed]

This is similar to the spherical pendulum. So much so that I'm wondering if there may be some typos?

[math]\ddot{ \theta } = sin( \theta) ~ cos( \theta ) ~ \dot{ \phi } ^2 - \dfrac{g}{l} sin( \theta )[/math]
[math]\ddot{ \phi } = -2 ~ cot( \theta ) ~ \dot{ \theta } ~ \dot{ \phi }[/math]
This system is similar to the ordinary one dimensional pendulum (set [math]\phi (t) = 0[/math]), which has no exact solution. This in turn implies that the spherical pendulum has no exact solution, either. Did you want to solve the case for both [math]\theta (t) \text{ and } \phi (t)[/math] are small?

-Dan

Yes, we do want to solve for the case of both theta and phi. Thank you for the diff eqn. corrections.

To progress, we need to re write our systems of differential equations in terms of subsitituted 1st order (linear?) ODEs so that we can numerically solve them. This is where I'm not sure as how to exactly do this.

 

[Deleted member 4993]

Yes, we do want to solve for the case of both theta and phi. Thank you for the diff eqn. corrections.

To progress, we need to re write our systems of differential equations in terms of subsitituted 1st order (linear?) ODEs so that we can numerically solve them. This is where I'm not sure as how to exactly do this.

As suggested in response #2:

Since you do not know where to start for these problems, please study the topic in the following web-site:

Differential Equations - Systems of Differential Equations

In this section we will look at some of the basics of systems of differential equations. We show how to convert a system of differential equations into matrix form. In addition, we show how to convert an nth order differential equation into a system of differential equations.

tutorial.math.lamar.edu

Did you study the contents of the web-site above? Since you do not have any work to show (still) - is there any specific part in there that you did not understand?

You say:

".......re write our systems of differential equations in terms of subsitituted 1st order (linear?) ODEs so that we can numerically solve them. This is where I'm not sure as how to exactly do this."

Did you study the example problems (1 - 4) worked out in that web-site? Those examples specifically address "re-writing".

 

[awaissssyed]

As suggested in response #2:

Did you study the contents of the web-site above? Since you do not have any work to show (still) - is there any specific part in there that you did not understand?

You say:

".......re write our systems of differential equations in terms of subsitituted 1st order (linear?) ODEs so that we can numerically solve them. This is where I'm not sure as how to exactly do this."

Did you study the example problems (1 - 4) worked out in that web-site? Those examples specifically address "re-writing".

Yes infact I studied that exact link before you suggested it. The problem is all of those examples show either one non linear DE in one variable or a system of DEs in 2 variables but they are linear, which in my case I have non linear, coupled, 2 DEs that are ub 2 variables, and no example online shows how to rewrite this.

 

[Deleted member 4993]

Yes infact I studied that exact link before you suggested it. The problem is all of those examples show either one non linear DE in one variable or a system of DEs in 2 variables but they are linear, which in my case I have non linear, coupled, 2 DEs that are ub 2 variables, and no example online shows how to rewrite this.

Why did you not indicate this with your original post?

Can you assume that

\(\displaystyle \theta << 1 \ \to \ sin(\theta ) = \theta \ \ and \ \ \cos(\theta ) = 1 \ \)