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# Separation of variables; PDEs watch

1. Hey guys, I'm having a bit of trouble understanding the working in my notes below (attatched picture just for context, but first line regarding X'' it corrcted below. I understand it until the point where it says that

when and ,

the 2 equations have the general solution:

Now I've started by putting or to account for being both positive or negative. From that I get 2 auxiliary equations: and each for both the X and Y equations.

However I can only achive
when

and
when

Can anyone help me understand where the other cases go?

ie.
when
and
when
2. (Original post by chris_d)

However I can only achive
when

and
when
Perhaps I'm being thick, but don't both those equations arise when you have , not from one postive and one minus.
3. (Original post by ghostwalker)
Perhaps I'm being thick, but don't both those equations arise when you have , not from one postive and one minus.
I agree with this.
4. (Original post by joostan)
I agree with this.
Thanks for the confirmation (PRSOM).
5. (Original post by ghostwalker)
Perhaps I'm being thick, but don't both those equations arise when you have , not from one postive and one minus.
I;m not really sure This is how much I understand this whole thing haha! These were my workings for the

let , therefore

then substituting

I'm not very good at all this...
6. (Original post by chris_d)
...
Let me just check something first:

I was assuming your alpha was real and the same in both equations. Is that the case?
7. (Original post by ghostwalker)
Let me just check something first:

I was assuming your alpha was real and the same in both equations. Is that the case?
In the case of the 2 equations, is equal in both of them, though it can be zero, real, imaginary or complex. I understand the case where , it's the case where I don't really understand. Sorry if I'm not making any sense
8. (Original post by chris_d)
In the case of the 2 equations, is equal in both of them, though it can be zero, real, imaginary or complex. I understand the case where , it's the case where I don't really understand. Sorry if I'm not making any sense
I'm not an expert on pds's, but I think it's highly unlikely that alpha is anything other than real in this context - but I could be wrong.

I am getting confused myself looking at your working. I can't see the error, but "know" there is one. Sorry.
9. (Original post by chris_d)
Hey guys, I'm having a bit of trouble understanding the working in my notes below (attatched picture just for context, but first line regarding X'' it corrcted below. I understand it until the point where it says that

when and ,

the 2 equations have the general solution:

Now I've started by putting or to account for being both positive or negative. From that I get 2 auxiliary equations: and each for both the X and Y equations.

However I can only achive
when

and
when

Can anyone help me understand where the other cases go?

ie.
when
and
when

please post a photo of the original PDE

separation of variables create F(x) = G(y)
therefore both are at most a constant positive, s2,
negative -s2
or zero

I need to see the PDE however
10. (Original post by ghostwalker)
I'm not an expert on pds's, but I think it's highly unlikely that alpha is anything other than real in this context - but I could be wrong.

I am getting confused myself looking at your working. I can't see the error, but "know" there is one. Sorry.
Aha, just like me then! I think I've reached a solution, though it's not one I'm very confident of. Here's the workings for X(x) (sorry it's so long) and I combined constants etc if thats ok?

for and so

and for where

and combining the 2 cases:

11. (Original post by chris_d)
.
I could well be being thick, and I've not done much on PDEs, but surely if is permitted to be complex then there is no necessity to consider the cases of ?
If you introduce this I feel that you are trying to end up with 4 linearly independent solutions for the CF, but there can only be two for each.
12. (Original post by TeeEm)
please post a photo of the original PDE

separation of variables create F(x) = G(y)
therefore both are at most a constant positive, s2,
negative -s2
or zero

I need to see the PDE however
Here's the pages in my notes regarding this The first equation is the 2D form of Laplace, and I'm looking at Example 1

13. (Original post by chris_d)
Here's the pages in my notes regarding this The first equation is the 2D form of Laplace, and I'm looking at Example 1

this is standard Laplace in solution in cartesian

what is the problem exactly?
14. (Original post by chris_d)
Here's the pages in my notes regarding this
I think tis isn't terribly well explained, and one thing is basically "wrong". He says can be imaginary but then talks about being positive or negative, which is essentially meaningless.

I've certainly never seen the sep. of vars. done where you'd allow the constant to be imaginary. (This isn't to say it can't work, but it's not what you'd normally do, and the 2nd page of your notes do actually seem to assume it is real);.

So, if it's real and non-zero, note that what you have is:

I think a lot of confusion has come up because of trying to distinguish between the alpha > 0 and alpha < 0 cases, because the way we'd naturally write the solution to has a different form for alpha > 0 and alpha < 0.

But it is in fact easy to verify that if alpha is non-zero, then

is the general solution to

(and similarly is the general solution to
).

Note that finding solutions in terms sin and cos and converting back into exponentials is basically a waste of time here. The sin/cos solutions are generally derived by finding the exponential solution and then combining to get real solutions. You're just undoing that again.

There's a symmetry here between X and Y;
15. (Original post by chris_d)
Here's the pages in my notes regarding this The first equation is the 2D form of Laplace, and I'm looking at Example 1

I hope you solved the issue wit the PDE

If this is not clear look at some more questions in the Laplace section of this link

you may want to also look at separation of variables for the wave equation or the diffussion/heat equation

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