Following are some course lecture notes written for the [MAT219] Introduction to Differential Equations course taught at METU Cyprus. The course uses Boyce and DiPrima's differential equations book, but is reorganized so that "Systems of Differential Equations" (Ch 7) is taught in the first part of the course -- before "High Order Linear" (Ch 3-4).

Since students were having trouble with "Systems of DE", I wrote up some (mostly) clean course notes for distribution and extra study. There are a few places with computational errors in these notes. I also included notes connecting to the "High Order Linear" chapters (3-4) because we organized our work there differently than Boyce and DiPrima. Also included are some extra notes reviewing Laplace transforms (Chapter 6).

The rough course outline is presented below. Sections without uploaded notes are written in strikeout.

• Course Summary Sheet.

• Directional Fields
• Separable Equations and "Homogeneous Separable" .
• Linear Equations
  Integrating Factors and Variation of Parameters
• Modeling
• Existence and Uniqueness Theorems
• Exact Equations .

• Introduction .
• Basic Theory .
• Basic Examples .
• Real and Complex Eigenvalue Cases .
• More Complex Eigenvalue Examples .
• Review of Systems so Far .
• Fundamental Matrix Notation (Matrix Exponentiation) .
• Generalized Eigenvectors (Jordan Form) Part I .
• Generalized Eigenvectors (Jordan Form) Part II .
• Variation of Parameters (2x2 and 3x3) .

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• Example Matrices with Real Eigenvalues .
• Example Matrices with Complex Eigenvalues .
• Example Matrices with Repeated Eigenvalues .
• Example Matrices for Jordan Decomposition .
• Side Notes: Computing Eigenvectors without Eigenvalues .

• Deriving Higher Order Linear from Systems (2 lectures).
• Undetermined Coefficients for Higher Order Linear .
• Variation of Parameters for 2nd Order Linear .
• Vibrations (see Course Summary Sheet)

• Laplace Transform Basics
• Step Functions and Differential Equations
• Impulse Functions and Differential Equations
• Convolutions and Green's Function (Impulse Response)

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• General Review of Laplace Chapter .
• Examples Computing Laplace Transforms .
• Examples Computing Inverse Transforms .
• Examples Computing Convolutions .

• Derivation of Heat Conduction Equation; Boundary Value Problems
• Fourier Series
• Fourier Series of Even and Odd Extensions
  (Sine and Cosine Series)
• Separation of Variables; Solution to Heat Equation

• Laplace Transform and Partial Differential Equations Review .