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