Applications of Differential Equations

This course introduces students to the theory of differential equations and dynamical systems and to their many applications as mathematical models. The topics covered prepare the student for more advanced subjects in ordinary differential equations (ODEs), dynamical systems theory, numerical methods and partial differential equations. Course Outline: A) 1st-order ODEs: Direction fields Linear equations: Integrating factor method. Separable equations. Exact equations. B) 2nd-order ODEs: Homogeneous equations: Characteristic equation method (real, complex and double roots). Non-homogeneous equations: Method of undetermined coefficients. Examples: Forced and damped systems, mechanical oscillations, resonances, etc. C) Systems of 2 coupled ODEs: Linear and nonlinear. Phase plane analysis. Critical points and classifications in terms of eigendirections and eigenvalues. Examples: Population models, the tragedy of the commons, epidemic models, etc.