Introduction to Differential Equations

• What is a differential equation?
• How do differential equations arrise?
• What are some of the models coming from differential equations?
• Solutions to differential equations
• Classification of differential equations

First Order Differential Equations

• Direction fields
• Euler's method
• The RK4 method
• Existence and uniqueness for first order differential equations
• Separable differential equations
• First order linear differential equations
• Exact differential equations
• Homogeneous first order differential equations
• Bernoulli equations
• Riccati equations
• The method of undetermined coefficients for first order

First Order Differential Equations Applications

• Radioactive decay
• Carbon dating
• Exponential growth
• Black hole evaporation
• Newton's law of cooling
• The logistic equation
• Terminal velocity
• RL circuits and RC circuits

Second Order Differential Equations

• Constant coefficient second order differential equations
• The method of undetermined coefficients for second order
• Euler-Cauchy equations
• Linear independence and the Wronskian
• Variation of parameters

Second Order Differential Equations Applications

• Mechanical vibrations
• Electronic circuits
• Displacement, velocity, and acceleration
• Beam displacement

Series Solutions

• Review of Power Series
• Series solution of differential equations
• Ordinary and singular points
• The method of Frobenius
• Special functions

Laplace Transforms

• Strategy of the Laplace Transform
• Definition of the Laplace Transform
• Mathematica and Laplace Transforms
• Laplace Transforms of some basic functions
• Properties of the Laplace Transform
• Inverse Laplace Transforms
• Laplace Transform of differential equations
• Step functions and delta functions
• Periodic forcing and Laplace Transforms
• Convolution Theorem
• Systems of differential equations with the Laplace Transform

Linear System of Differential Equations

• Definition and classification of linear systems
• The matrix exponential
• Eigenvalue eigenvector methods
• Two by two linear systems and trace determinant plane
• Nonlinear systems of differential equations
• Predator-prey systems
• Linearization and local behavior