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# The traditional approach to ordinary differential equations

• introduces analytic solutions of differential equations, but
• does not cover the qualitative analysis needed for intuition, or
• numerical methods needed for applications
• linear algebra is introduced very late in the course, and is not covered deeply.

## What is new?

• When teaching analytic methods for first-order ODEs, we also introduce slope fields, phase lines, stable and unstable equilibria and bifurcations.

• We link linear equations with harmonic oscillation, separation of variables with population models, and exact equations with thermodynamics.

• After first-order ODEs, we introduce linear algebra -- Gaussian elimination, matrix operations, eigenvalues and eigenvectors -- and apply them to systems of first-order linear ODEs.

• We emphasize that the structure of solutions to linear systems applies not only to systems of linear ODEs but also to systems of linear algebraic equations and systems of linear integral equations.

• Phase planes, qualitative analysis of linear systems and stability are introduced after analytic solutions and enhanced with MATLAB and MAPLE demonstrations and assignments.

• The operator method and Laplace transforms are covered. Then we proceed to systems of nonlinear differential equations, including linearization, Hopf bifurcation, limit cycles, Lorentz equations, and chaos in chemical systems.

• To better understand chaos, we also introduce chaos in discrete systems.

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