Many laws that govern the natural and technological world are relations (equations) involving rates (derivatives). Equations containing derivatives are ... differential equations.
Some are. A great many are not.
Solutions are functions. If they cannot be expressed symbolically, they are mathematical formulas. Otherwise they are curves, or even tables of numbers.
Probably not. Too many equations, too little time. What the course can do is help you to become familiar with some powerful methods and tools that you can use to investigate many kinds of differential equations.
Some can be carried out with pencil and paper. Others require a computer for efficient implementation.
Yes. You will need to take advantage of a computer analysis system like Maple frequently.
First is Maple's ability to draw graphs of solutions, which often makes their important features apparent and which usually are very time-consuming to draw without a computer. Second is Maple's symbolic differential-equations solver that produces formulas in most of the cases where such expressions are known to exist. Third is Maple's help in executing the many calculations that are needed to extract information from solutions.
A key question. Sometimes you will be told, but one of the course goals is for you to learn and to decide for yourself when it can be helpful.