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An ordinary differential equation (ODE) is an equation relating the derivative of a function y of some variable t to the function y itself and to the variable t. When t is time, this means that the ODE describes the time-rate of change of the quantity y in terms of the quantity and the time. Since so many principles describing the world involve rates of change, ODEs pervade engineering, the sciences, and the more quantitative social sciences and management sciences.
The overall goal of this course is to say as much as possible about solutions of ODEs by studying ODEs in three ways:
- analytically (symbolically), that is, by finding exact formulas for the solutions;
- qualitatively, that is, by determining the behavior of the solutions (even when exact formulas cannot be found); and
- numerically, that is, by constructing approximations to solutions (especially when exact formulas for solutions cannot be found).
The more specific learning goals for the course are:
- To apply basic paper-and-pencil skills of solving ODEs analytically.
- To use graphical and other means to describe the qualitative behavior of solutions of ODEs without necessarily finding the solutions analytically.
- To use numerical methods to approximate solutions of ODEs.
- To apply matrix methods in studying and solving ODEs.
- To understand how complex numbers arise from, and are applied to, ODEs.
- To know when and how to use a calculator or computer software (instead of paper-and-pencil calculation) to solve problems about ODEs, and to know which calculator or computer modalities—symbolic, numeric, and graphical—are appropriate.
- to present clearly and coherently written solutions to problems with mathematical content.
See also the list of topics on the Content page.
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