Variation to the Helmholtz equation

I was successful @ splitting the coupled eq into the 3 constituent PDE's but have been unsuccessful to get MAPLE to generate the characteristic solutions except for the 1st one for X which is the basic solution to the Helmholtz eq.

Can someone with more savy with MAPLE verify whar I have so far & advise me how to proceed to solve for Y & Z?  I have uploaded my file.  My expereince with MAPLE vector calc package is limited.

appreciate the assistance