What topics should I review for self-studying differential equations?

[aspiring_one]

Hello,

I was wondering what topics I should consider brushing up on in order to self-study differential equations. I had to take a semester with no math class but I'm still competent in algebra, trig, and some concepts of calculus (up to multivariable). Are there any topics in calculus or maths in general that I should specifically review?

Thanks

 

[Fizex]

Differentiation, integration techniques, and linear algebra.

 

[aspiring_one]

Differentiation, integration techniques, and linear algebra.

Thanks, what sorts of multivariable calc should I know or seem useful?

 

[nlsherrill]

Thanks, what sorts of multivariable calc should I know or seem useful?

just partial derivatives. There aren't many multivariable calc topics on ODE's. Maybe brush up on your vectors though.

 

[Angry Citizen]

Also brush up on your power series.

 

[aspiring_one]

Thank you all for your replies. Should I pay any attention to graphs and max's and mins? that type of stuff. Can you graph a diff eq? also shall i bother with the shell,washer,and disc methods? what about reimann sums, the trapezoid rule, and simpsons rule for approximating area?

Thanks again, I really want to put my effort on the right topics.

 

[flyingpig]

I am self-studying ODEs right now and the important concepts you need from Calculus are

Chain Rule of Derivatives, separable functions, euler's method/direction fields, a little bit of limits and just maybe improper integrals. Also chain rules in partial derivatives

 

[Leveret]

You can't typically "graph a diff eq" per se, because most differential equations have an infinite number of solutions, depending on your initial and boundary conditions. A simple example would be y'(x)=y(x), which could mean y=e^x, y=2e^x, or y=3e^x, etc. What you can do is draw a solution set, which looks kind of like a contour map, showing a representative sample of the various possible solutions. Insofar as you're concerned, though, this means optimization (finding maxes/mins) plays a relatively small role in ODEs, although it does have some applications in PDEs.

The shell, washer, and disc methods aren't used much in ODEs either. However, they certainly could crop up in some applications, as they and ODEs are both widely-applicable techniques. You can probably get through ODEs without them, but you should know them anyway.

Riemann sums, the trapezoidal rule, and the like will probably show up if you study any numerical methods of solving differential equations. I wouldn't bother learning all the nitty gritty details, though: just have a feel for the general philosophy behind them, and you should be ok.

 

[osnarf]

You can graph an ODE, it represents a vector field so it won't look like a regular function graph, per se. This website gives a quick overview of vector fields, I didn't read through but look at the pictures, it should ring a bell from calc 3:

http://www.sosmath.com/diffeq/slope/slope1.html

Arnold (V.I. Arnold - "Ordinary Differential Equations") says a differential equation IS a vector field. His approach is very geometric. I started to read through it and got a good ways in but detoured to read calculus on manifolds (more rigorous multivariable calc) so I could get more out of arnold's book. However, the beginning parts of the book are not too mathematically intense and will give a very good idea of what differential equations are and what vector fields are, and I really recommend reading it (the beginning anyways - I'm really looking forward to getting back to the rest of it, but it's definitely higher level).

If what you're taking is an intro class to differential equations in college - IE the class that usually comes after the regular calculus sequence - I really wouldn't fret about it too much. You will probably find it to be much less challenging then calc 2 or 3. Some specific things you may want to make sure you know: product rule, integration techniques like integration by parts (single variable), eigenvalues if your class is going to go over systems of DE's... and that's really it. The class could probably be summed up on 3 peices of paper - it's mostly recognizing certain types of differential equations and solving them with cut and paste methods.

 

[osnarf]

Also, if you haven't studied differential equations before, you're probably not going to encounter PDE's in your first class/book.

 

[Angry Citizen]

That's not accurate. My DE class will cover some PDE's, but to what depth I'm not certain.

 

[aspiring_one]

Thanks for the input osnarf. Vector fields sound sort of similar to gradients but I am not sure. Anyway the course is supposed to cover ODE's, PDE's and Fourier series. I really understood calculus through a geometrical and physical approach so I'll look into that book, hopefully my library has it.

 

[osnarf]

Lol mine didn't go anywhere near pde's, that's in emath, sorry for assuming it was the same elsewhere.

OP: if you do read it, after a little ways in it starts to get pretty hard, i only meant to read the beginning before it gets too technical. Like I said I had to put it down and go learn more first so don't get discouraged once you get to the point in talking about, its not you, it's meant for people who already know a good bit about the material. If you do get it Bravo lol.