Students normally study the topic of differential equations after several semesters of calculus, at which time it becomes a natural extension of the former. Simply defined, differential equations are equations that contain one or more differentials of functions. A course in differential equations will be focused on techniques for solving the various types of these equations, with particular focus on those that are so important in math and science.
One example of the use of differential equations is in the field of optical physics, where electrons in dielectric materials are modeled by a negative charge suspended by springs, free to oscillate in the presence of time-varying electric and magnetic fields, such as those in an ordinary light wave. The electron's equation of motion can be easily described with a differential equation whose solution leads to an expression for the index of refraction as a function of the frequency of the incoming light wave. It is this development of the index of refraction that leads to a complex representation for the index of refraction that prevents the index from becoming discontinuous during absorbtion of light by the dielectric.
A good course first course in differential equations will involve the following:
- First-order differential equations
- First-order systems
- Linear Systems
- Forcing and resonance
- Nonlinear systems
- Laplace transforms
- Numerical methods
- Discrete dynamical systems
Many good books on differential equations are available on Amazon.com. You can also get a free differential equations ebook online. Don't miss a solid differential equations tutorial at the Scribd website.
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