Using separation of variables
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.
...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).
Think about it like this. You have the two solutions,
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.