Using separation of variables

Now what?

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- January 1st 2011, 06:29 PMdwsmithu_{xy}+u_{yz}+u_{zx}-u=0
Using separation of variables

Now what? - January 1st 2011, 07:04 PMAckbeet
Well, pick a constant of separation:

The first equality gives you a DE for The second equality gives you another equation that, I believe separates out thus:

You can do the same sort of trick you just did. That is, solve for and choose another separation constant.

You follow? - January 1st 2011, 09:25 PMchiph588@
- January 1st 2011, 09:36 PMdwsmith

I tried the substitution of in order to manipulate the exponential method to the separations method but it fell short.

Exponential solution:

What substitution do I need to make?

Thanks. - January 1st 2011, 09:40 PMdwsmith
This is from Elementary PDE by Berg and McGregor 1966.

...different technique for finding a particular solutions of homogeneous linear PDE is called the method of separation of variables. The exponential solution , are products of functions of the separable variables.... We seek a solution of the form (pg. 14). - January 1st 2011, 11:19 PMSudharaka
Dear dwsmith,

Think about it like this. You have the two solutions,

--------(Seperation method)

---------(Exponential method)

Consider the exponential part with y as the variable. Since the coefficient of y must be equal in both of these equations,

Now move on to the exponential term with z as the variable. Using the same reasoning,

(1) and (2) gives the substitutions necessary to convert the solution obtained from the seperation method to the solution obtained from the exponential method. Hope you understood. - January 2nd 2011, 01:40 PMdwsmith
By making that substitution, we don't obtain the same solutions.

Any thoughts?

I could say but would this be ok? - January 2nd 2011, 05:38 PMSudharaka
- January 2nd 2011, 05:42 PMdwsmith