Things I must know before taking differential equations course

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.

I think the best thing you could do at this point is to pick up Schaum's Outline of Calculus or your old textbook, if you still have it, and begin working problems to refresh your memory. You will need to know all of the basics of calculs including differentiation, integration, series, methods of approximation and so on to do well in DiffyQ. Since you've been away from math for awhile, it also wouldn't hurt you to review basic algebra/trig before you begin reviewing calculus. Depending on how much time/day you plan to devote, you might want to allocate a month or so for a thorough review.