This page is directed at graduate students in UCLA's Department of Mathematics preparing to take qualifying exams. The best way to study for any qual at UCLA is to practice on old qual questions, as themes and even entire questions get repeated. (The Basic Exam is no exception to this, by the way.) As you go through old qual questions, remember that occasionally the powers that be have written questions that are impossible by hand in four hours, or impossible without additional assumptions beyond what's stated, or impossible because they ask you to prove a falsehood. I took the "applied" quals (Numerical Analysis and ADEs) so from here on I'll restrict my comments to those.
You need to do well on both the "undergraduate" and "graduate" sections to pass. You should be able to handle the ODE and finite difference questions after taking 269A and B, respectively, although I would recommend reading the section on symbols in Strikwerda. Symbols are not generally taught in 269B, but they can save you precious time on the qual.
If you haven't had undergraduate numerical analysis, buy or borrow a copy of Burden and Faires, the textbook used in UCLA's undergrad numerical class, and read through the sections covered in the qual syllabus. Pay particular attention to root solving, numerical differentiation, numerical integration, and linear algebra (Gaussian elimination, Jacobi, Gauss-Seidel, etc.), as these topics seem to come up most often in the "undergraduate" section. That said, be prepared for anything.
The finite element question should be manageable if you've had 269C. If you haven't, buy or borrow a copy of Claes Johnson's Numerical Solution of Partial Differential Equations by the Finite Element Method. (If you plan to take 269C, you will need it anyway.) Prof. Luminita Vese has made available a good list of useful results and a couple practice problems with partial solutions.
Bryon Aragam and Alejandro Cantarero have assembled some relevant notes and solutions to old qual problems. Pascal Getreuer and Jeff Hellrung also had helpful notes back when they were students here; maybe you can track them down.
Applied Differential Equations
In 266A, pay particular attention to phase plane portraiture, Lyapunov's direct method, and boundary-value problems. You will need pretty much everything you learn in 266B. It's helpful to be comfortable with Fourier theory, Sobolev spaces, and other analysis topics that are not necessarily covered in detail in 266A and B. Prof. Vese's list of useful results, though nominally intended for finite element questions on the Numerical qual, is useful for ADEs, as well.
I've posted my solutions to some old ADEs questions. Other sources for notes and practice problems are the websites of Bryon Aragam, Alejandro Cantarero, Lee Ricketson, and Igor Yanovsky. Pascal Getreuer and Jeff Hellrung also had some.