Separation of variables and the separation constant

[Niles]

Homework Statement

Please take a look at:

http://www-solar.mcs.st-and.ac.uk/~alan/MT2003/PDE/node21.html

Look at step 2.53. Can you explain to me how c^2*T'/T = k becomes T' = k*T*c^2?

The Attempt at a Solution

I don't get it. What am I missing here?

 

[tiny-tim]

Yes, you're right!

Hi Niles!

Yes, you're right!

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

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

(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.)

 

[Niles]

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.

 

[tiny-tim]

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!

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?

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

 

[Niles]

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

(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.