Free Differential Equations and Linear Algebra Notes

Teaching

I have a license to teach mathematics at the secondary level in the state of Minnesota.

Minnesota State Teaching License

As an instructor for Differential Equations and Linear Algebra, I prepared these class notes as a review of the sections in Edwards and Penney's Differential Equations and Linear Algebra . These notes include worked examples, exercises, and solutions on differential equations, matrices, vector spaces, eigenvalues, systems of differential equations, phase planes, Laplace transforms, and power series methods.

Use the sections below to find materials by topic. Each unit can be expanded or collapsed to make the page easier to browse.

First-Order Differential Equations

Introductory differential equation models, slope fields, separable equations, linear first-order equations, substitutions, and exact equations.

Class NotesIn-Class ExercisesSolutions
1.1 Differential Equations and Mathematical ModelsExercises 1.1Solutions 1.1
1.2 Integrals as General and Particular SolutionsExercises 1.2Solutions 1.2
1.3 Slope Fields and Solution CurvesExercises 1.3Solutions 1.3
1.4 Separable EquationsExercises 1.4Solutions 1.4
1.5 Linear First-Order EquationsExercises 1.5Solutions 1.5
1.6 Substitution Method, Exact EquationsExercises 1.6Solutions 1.6
Numerical and Qualitative Methods

Mathematical models, equilibrium solutions, stability, velocity models, Euler's method, and improved Euler methods.

Class NotesIn-Class ExercisesSolutions
2.1 Mathematical Models and Numerical MethodsExercises 2.1Solutions 2.1
2.2 Equilibrium Solutions and StabilityExercises 2.2Solutions 2.2
2.3 Acceleration Velocity Models——
2.4 Euler MethodExercises 2.4Solutions 2.4
2.5 Advanced Euler Method——
Linear Systems and Matrices

Linear systems, reduced echelon form, matrix operations, inverses, determinants, and curve fitting.

Class NotesIn-Class ExercisesSolutions
3.1 & 3.2 Linear SystemsExercises 3.1
Exercises 3.2		Solutions 3.1
Solutions 3.2
3.3 Reduced Echelon MatricesExercises 3.3Solutions 3.3
3.4 Matrix OperationsExercises 3.4Solutions 3.4
3.5 Inverses of MatricesExercises 3.5Solutions 3.5
3.6 DeterminantsExercises 3.6Solutions 3.6
3.7 Linear Equations and Curve Fitting——
Vector Spaces

Vector spaces, subspaces, linear combinations, independence, bases, dimension, row and column spaces, and general vector spaces.

Class NotesIn-Class ExercisesSolutions
4.1 Vector SpaceExercises 4.1Solutions 4.1
4.2 SubspacesExercises 4.2Solutions 4.2
4.3 Linear Combinations and Independence of VectorsExercises 4.3Solutions 4.3
4.4 Bases and Dimension for Vector SpacesExercises 4.4Solutions 4.4
4.5 Row and Column Spaces——
4.7 General Vector Spaces——
Second-Order Differential Equations

Second-order linear equations, homogeneous and nonhomogeneous equations, mechanical vibrations, forced oscillations, and resonance.

Class NotesIn-Class ExercisesSolutions
5.1 Second-Order Linear EquationsExercises 5.1Solutions 5.1
5.2 General Solutions of Linear EquationsExercises 5.2Solutions 5.2
5.3 Homogeneous Equations with Constant CoefficientsExercises 5.3Solutions 5.3
5.4 Mechanical Vibrations——
5.5 Nonhomogeneous Equations and Undetermined CoefficientsExercises 5.5Solutions 5.5
5.6 Forced Oscillations and Resonance——
8.2 Nonhomogeneous Linear Differential EquationsExercises 8.2Solutions 8.2
Eigenvalues, Diagonalization, and Matrix Powers

Eigenvalues, diagonalization, and applications involving powers of matrices.

Class NotesIn-Class ExercisesSolutions
6.1 EigenvaluesExercises 6.1Solutions 6.1
6.2 Diagonalization of MatricesExercises 6.2Solutions 6.2
6.3 Powers of MatricesExercises 6.3Solutions 6.3
Systems of Differential Equations

First-order systems, matrices and linear systems, the eigenvalue method, solution curves, mechanical applications, multiple eigenvalues, and numerical methods for systems.

Class NotesIn-Class ExercisesSolutions
7.1 First-Order SystemsExercises 7.1Solutions 7.1
7.2 Matrices and Linear SystemsExercises 7.2Solutions 7.2
7.3 Eigenvalue Method for Linear SystemsExercises 7.3Solutions 7.3
7.4 Solution Curves of Linear SystemsExercises 7.4Solutions 7.4
7.5 Second Order Systems and Mechanical Applications——
7.6 Multiple Eigenvalue——
7.7 Numerical Methods for Systems——
Phase Plane and Stability

Stability, phase plane analysis, linear and almost-linear systems, and ecological models.

Class NotesIn-Class ExercisesSolutions
9.1 Stability and the Phase PlaneExercises 9.1Solutions 9.1
9.2 Linear and Almost Linear SystemsExercises 9.2Solutions 9.2
9.3 Ecological Models——
Laplace Transforms

Laplace transform methods, initial value problems, translation, and partial fractions.

Class NotesIn-Class ExercisesSolutions
10.1 Laplace TransformExercises 10.1Solutions 10.1
10.2 Laplace Transformation of Initial Value ProblemsExercises 10.2Solutions 10.2
10.3 Translation and Partial Fractions——
Power Series Methods

Power series methods for solving differential equations.

Class NotesIn-Class ExercisesSolutions
11.1 Power Series MethodExercises 11.1Solutions 11.1