Separation of variables and the separation constant

In summary, Niles is asking about why the c^2 is included in equations (15) and (18) when it doesn't need to be, and why the separation constant is -omega^2/c^2 instead of just omega. He thinks that perhaps it is done for dimensional reasons. Tim thinks that (16) to (18) look better with the c^2 included, and that it is always a good thing to include the constant in order to get T''/T = -omega^2.
  • #1
Niles
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  • #2
Yes, you're right!

Hi Niles! :smile:

Yes, you're right! :smile:

2.5.3 is correct, but the c^2 in the following line should be 1/c^2. :frown:

But it doesn't matter, because the c^2 in 2.5.5 is correct, since it's taken directly from 2.5.3 :smile:

(The line after 2.5.3 was only used to get the sign of k, so it didn't matter whether k was multiplied or divided by c^2.)
 
  • #3
Hi Tim, thanks for replying. I hope it's OK if I ask another question.

Please take a look at http://cow.physics.wisc.edu/~craigm/toroid/toroid/node4.html

Here, they choose the separation constant to be -omega^2/c^2. What is the deal when finding the separation constant (SC from now on)? In my book they equal the exact same term to -omega^2. Should I include the constant in front of T''/T every time when choosing SC?

Thanks in advance.

Sincerely Niles.
 
  • #4
Niles said:
Here, they choose the separation constant to be -omega^2/c^2. What is the deal when finding the separation constant (SC from now on)? In my book they equal the exact same term to -omega^2. Should I include the constant in front of T''/T every time when choosing SC?

Hi Niles! :smile:

The c^2 has to be in equation (13) because (13) is taken directly from (11).

The c^2 in equations (15) and (18) doesn't have to be there - omega is any constant, so it doesn't matter whether you choose omega or omega/c.

I think he's done it that way for dimensional reasons. I'm inclined to agree with him.

Don't you agree that (16) to (18) look much neater than they would if (18) included c^2? :smile:

(Does your book use c, or does it put c = 1?)
 
  • #5
tiny-tim said:
Don't you agree that (16) to (18) look much neater than they would if (18) included c^2? :smile:

(Does your book use c, or does it put c = 1?)

I agree - it does look better. So it is always a good thing to include the constant so we end up with a term T''/T = -omega^2?

My book uses c. It is the exact same eq. as in the link.
 

1. What is separation of variables and how does it relate to the separation constant?

Separation of variables is a mathematical technique used to solve differential equations. It involves separating a multivariable function into simpler functions with only one variable. The separation constant is a constant that appears in the separated equations and helps to solve the original equation by providing additional information about the solution.

2. Why is the separation constant important in solving differential equations?

The separation constant provides an additional constraint to the separated equations, making it easier to solve for the variables. It also helps to ensure that the solution satisfies the original differential equation.

3. Can separation of variables be used to solve any type of differential equation?

No, separation of variables can only be used to solve certain types of differential equations, specifically those that are separable and have two or more variables. It is not applicable to all types of differential equations.

4. What are the steps involved in using separation of variables to solve a differential equation?

The first step is to identify if the equation is separable and has two or more variables. Then, the variables are separated by moving all terms containing one variable to one side of the equation and all terms containing the other variable to the other side. Next, the separated equations are solved individually, with the separation constant included. Finally, the solution is combined using the separation constant to obtain the final solution.

5. Are there any limitations or drawbacks to using separation of variables to solve differential equations?

Yes, separation of variables can only be used for specific types of differential equations and may not always result in a closed-form solution. It also requires a certain level of mathematical understanding and may not be applicable to all real-world problems. Additionally, the separation constant may not always be easy to determine, making the solution process more complex.

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