Why Study Differential Equations?

• Why should one be interested in differential equations?
  
  Many laws governing natural phenomena, finance, engineering, etc. are relations (equations) involving rates at which things happen (derivatives). Equations containing derivatives are differential equations
  
  
• So to be able to investigate realistic problems in fluid mechanics, circuit design, control systems, heat transfer, structure of beams, population or conservation biology, neurobiology, seismic waves, economics, derivative trading,..., I need to know something about differential equations? In fact, you may have been studying math your whole life to get to this course.
  
  Right
  
  
• Are differential equations easy to solve?
  
  Some are, but many are not
  
  
• What do solutions look like?
  
  Solutions are functions, so if expressed symbolically they look like mathematical formulas. Geometrically, they are curves.
  
  
• Will I learn in this course how to solve all the differential equations that I will ever be interested in?
  
  Probably not. It is hard to anticipate which equations you might want to solve in the future. In addition, there is not enough time to study all types of equations.

  What we can do is help you become familiar with some powerful methods and tools that can help you investigate many kinds of differential equations.
  
  

• What kinds of methods and tools?
  
  That's what the course is about, so you will see them as we go along. Some require hand or SYMBOLIC manipulations, other are based on NUMERICAL computations. Often it is desirable to present results in GRAPHICAL form, so that the behavior of solutions can be easily visualized.
  
  
• How often will I need to use a computer in this course?
  
  We will expect you to use a computer frequently, not only to obtain course information from the Differential Equation Homepage but primarily to run Maple. Sometimes a pocket calculator may be helpful. But many things are done better by hand with paper and pencil. What we are looking for is BALANCE. We want you to leave this course with a broad view of differential equations, their possible applications, and with confidence in using a wide variety of methods.
  
  
• Why is Maple useful in the study of differential equations?
  
  There are several reasons. Perhaps most important is Maple's ability to draw graphs of solutions, which often makes their important features much more apparent. Maple also has a powerful symbolic differential equations solver that produces expressions for solutions in most cases where such expressions are known to exist. Maple can also be used to carry out numerical calculations on differential equations that cannot be solved in terms of simple expressions.
  
  
• How will I know when a computer is needed?
  
  Sometimes we will tell you, other times you will have to decide for yourself whether it will be helpful to use a computer

Adapted from Diff Eq course at RPI