MHB Problem 13 from section 16.1 of Taylor's PDE textbook.

[Alone]

I was given as a task to solve this question by my teacher (heck if I had the time I would have solved every problem in both Taylor's and Evans's books on PDE); but didn't succeed to the teacher's satisfaction.

In the following link there's a presentation of the problem, and in the attachment my failed attempt at solving it (it should have been part of my thesis I guess :-( ):
https://math.stackexchange.com/questions/1180968/question-13-in-taylors-pde-vol-iii-section-16-1

Any help on how to solve this?
In the attachment there's the latex file with the remarks my teacher gave me. (His remarks are between the two lines with*******).

Perhaps one day I'll get my MSc...
Taylor_Exercises-new

I tried to upload as an attachment through the option in the MHB forum but didn't succeed as the txt file exceeds the site's capacity, so I uploaded to mediafire.

Thanks in advance for your time!

 

[Alone]

I am attaching a compiled PDF of the above TEX file.
View attachment 7470