The final is NOT comprehensive. It covers all of Chapter 5, except 5.7, and covers 7.5, 7.6, and the notes on Exponential Series on the web page. We also studied reduction of order and the Wronskian. (See the exercises for 4/6 and 4/8.) In general, you should study all of the homework.
This semester we did not cover Laplace Transforms. I thought that it was more useful to spend additional time on series solutions. Hence, you should ignore problems (11)-(15) on last fall's final and problems (11)-(14) on last spring's final. Instead, I will probably ask more questions on other topics such as series solutions. For example I might ask a question similar to questions (7) and (8) on the final from last spring where your substitution involves x^(n+r) instead of x^n.