Topics to master (be literate at) before differential equations? Good evening, I'm really enthusiastic about learning differential equations because it was said that D.E. is the most important tool of mathematics "can be used for modelling real-world physical occurrences". 
I've taken courses in Differential and Integral Calculus (including numerical techniques of evaluation), and had self-studied Multivariable Calculus (only from partial differentiation, multiple integration, Vector Integration Stoke's, Green's), Series (taylor/maclaurin, covergence, divergence), and a little bit of linear algebra (before vector spaces). 
I have a book on Differential Equations but it seems that I can't understand the way it proves and explains the rules and theorems (I can solve separable first order and the one with dy/dx + y = c, but I'm stuck after it, specifically on the uniqueness and existence of solutions).
So for the people who are adept at D.E., what mathematical techniques/knowledge should I study to further understand Differential Equations? (My plan now is to finish my linear algebra book)
PS. I am an engineering major :), though my math courses are focused on the application and problem solving, I keep it to a point that I know its basis and that I can derive it.
 A: Your plan is very good. You absolutely need to finish at least an introductory linear algebra book before seriously dealing with differential equations.
By seriously, I mean "in a way that you know what the heck you are doing". No offense to engineers and physicists, but they start DEs way to soon and before they are capable of understanding what is happening. I remember when studying mathematics, I was the only one of my high school friends in math-heavy studies that only started differential equations in my third year of studies. Physics students encountered them in their third week, far before they had any theoretical knowledge to understand what was happening (so to many of them, DEs still exist as some voo-doo space where hand waving and strange leaps of logic lead you to the correct result).
For any serious work on diferential equations (and a lot of other parts of mathematics), you need more knowledge of linear algebra.
From linear algebra, you need to cover the topics:


*

*Vector spaces

*Bases and changes of bases

*Linear mappings

*Matrices and their relation to linear mappings

*Eigenvalues and eigenvectors

*Characteristic and minimal polynomial of a matrix

*Jordan canonical form


Furthermore, for uniqueness of solutions, you need some introduction into metric and normed spaces. You need knowledge of


*

*Metrics

*Sequences and limits in metric spaces

*Normed spaces, maybe even inner-product spaces (related also to linear algebra)

*Complete metric spaces

*The banach contraction principle (this one is crutial!)

A: [I have a different take in that linear algebra is NOT required BEFORE a FIRST course in ODE.  Not to be contentious, but I think you need to have the other side of the story on this Q/A.  I am not surprised to hear that advice on this forum (with a math purist slant) but you should really consider some alternate viewpoints.]  
Math is something that is iterative and branching and recursive.  You would never have learned addition in elementary school if you had to start by defining the number systems.  Just because you don't take a full course in LA now, doesn't prevent you from taking one later.  And for many courses, it is more effective to learn things first simply and then go deeper, rather than to go deep from the start (this is the "calculus versus real analysis debate").  
Finally, I am not the end-all, be-all of engineers but have taken EIT and PE (mechanical) and worked in an ME/EE firm and also have Ph.D. in a hard science.  My LA experience consists of the smattering that you get from a few weeks in "engineering math" (after ODE) and even there, it was very short since there was more time spent on PDEs.  I do have a book on LA and sampled a course (dropped from time requirements).  Bottom line:  would I like to know some more LA?  Sure.  Has it hurt me even in hairy courses like grad superconductors?  Nope. (Now the darned Bessel functions...those darned things turn up everywhere!)


*

*I have 4 standard diffyQ books on my shelf and none of them require linear algebra as a prerequisite. To the extent that LA concepts are touched on (a SMALL part of the ODE course and more advanced parts...systems, Sturm Louisville...often skipped in a first course that is semester long) these topics are handled fine by the books.  If you need a determinant (and not that often in the course you need one) they show you what a determinant is...big deal.

*I have attended multiple universities and none of them had linear algebra as a prerequisite course for diffyQs.  This is NOT the norm that you need LA as a prereq.

*I have 4 popular PDE books on my shelf.  None require LA as a prereq either.  All teach the (limited) LA needed as it is needed.

Bottom line:  you are getting a distorted, not conventional wisdom view, with the idea that you need to master LA before taking first course in ODE.  You will do fine in ODE without that LA course.  And if you choose to take it later, nothing stopping you.  [And the uses of LA are really more in topics different than ODE, PDE, engineering modeling. Linear programming, advanced stats, etc.]
