Professor's Notes
These lecture notes are provided for students in MAT225 Differential Equations. The are based on the presentation in Boyce and DiPrima. They guide but do not necessarily correspond exactly to the classroom presentation.
- 1.1 Some Basic Mathematical Models; Direction Fields
- 1.2 Solutions of Some Differential Equations
- 1.3 Classification of Differential Equations
- 2.1 Linear Equations with Variable Coefficients
- 2.2 Separable Equations (Justification)
- 2.3 Modeling with First Order Equations
- 2.4 Differences between Linear and Nonlinear Equations
- 2.5 Autonomous Equations and Population Dynamics
- 2.6 Exact Equations and Integrating Factors
- 2.7 Numerical Approximations: Euler's Method
- 3.1 Homogeneous Differential Equations with Constant Coefficients
- 3.2 Solutions of Linear Homogeneous Differential Eqs.; the Wronskian
- 3.3 Complex Roots of the Characteristic Equation
- 3.4 Repeated Roots; Reduction of Order
- 3.5 Nonhomogeneous Differential Eqs: Undetermined Coefficients
- 3.6 Variation of Parameters
- 4.1 General Theory of nth Order Linear Differential Equations
- The n Roots of a Complex Number
- 4.2 Homogeneous Differential Equations with Constant Coefficients
- 4.3 The Method of Undetermined Coefficients
- 5.1 Review of Power Series
- 5.2 Series Solution near an Ordinary Point, Part I
- 5.3 Series Solution near an Ordinary Point, Part II
- 5.4 Euler Equations; Regular Singular Points
- 5.5 Series Solutions near a Regular Singular Point
- 6.1 Definition of Laplace Transform
- 6.2 Solution of Initial Value Problems
- 6.3 Step Functions
- 6.4 Differential Equations with Discontinuous Forcing Functions
- 6.5 Impulse Functions
- 7.1 Introduction to Systems of 1st Order Differential Equations
- 7.2 Review of Matrices
- 7.3 Systems of Equations, Eigenvalues and Eigenvectors
- 7.4 Theory of 1st Order Systems
- 7.5 Homogeneous Linear Systems with Constant Coefficients
- 7.6 Complex Eigenvalues
- 7.7 Fundamental Matrices
- 7.8 Repeated Eigenvalues
- 9.1 The Phase Plane: Linear Systems
- 9.2 Autonomous Systems and Stability
- 9.3 Locally Linear Systems
- 9.4 Competing Species
- 9.5 Predator-Prey Equations