[Math] Things I must know before taking differential equations course

ordinary differential equationsreference-request

I intend to take this course named "Differential Equations" and per the department followings contents will be taught

*  First Order Differential Equations
* Second Order Linear Equations
* Series Solutions of Second Order Linear Equations
* Higher Order Linear Equations
* The Laplace Transform
* System of First Order Linear Equations
* Partial Differential Equations and Fourier Series
* Boundary Value Problems and Sturm Liouville Theory
* Non Linear Differential Equations

and this is also given in the course outline

"
In this course the students will learn how to solve boundary value problems analytically. This will enable them to develop command over one of the two techniques, namely:
– Analytical Techniques
– Numerical Techniques for the solution of boundary value problems
"

Now I've not taken math course in a while (10 years ago I took Calculus in high school and I don't remember most of it unfortunately).

So I've two questions that I hope some of you can answer for me

  • What topics in Calculus I must know before taking this course?
  • What is the best Differential Equations book for person like me given the above course outline?

Please pardon my ignorance. I will really appreciate all the help. Thanks and I look forward to hearing from you.

Best Answer

The best thing for you to do would be to look at the prerequisites of the course as specified by your University. Only they will be able to tell you what knowledge you should have before taking the course. From the description give, the course could be at first year undergraduate level, or it could be at graduate student level.

However, I will have a stab. This is, and necessarily must be, an incomplete list:

  • You should have facility with the calculus of basic functions, eg $x^n$, $\exp x$, $\log x$, trigonometric and hyperbolic functions, including derivatives and definite and indefinite integration
  • The chain rule, product rule, integration by parts
  • Taylor series and series expansions
  • Differentiation from first principles, as the limit of ratio of differences
  • Riemann integrals
  • Linear algebra at the level where you're comfortable with the notions of a linear transformation, representing a linear transformation as a matrix, eigenvalues and eigenvectors, and matrix inverse
  • Complex numbers, including cartesian and polar representation, Euler's formula, and relations with trigonometric and hyperbolic functions

Others should feel free to edit with anything I've left out.

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