Notes on linear algebra and ODEs

Applied Math for Engineers

I. Linear algebra

1. Overview
2. Vectors and matrices
3. Linear systems
4. Row elimination
5. RREF
6. Vector algebra
7. Matrix-vector algebra
8. Matrix multiplication
9. Identity and inverse
10. Detecting singularity
11. General solutions
12. Complex numbers
13. Eigenvalues
14. Multiplicity

II. First-order linear ODE

1. Overview
2. Quick tour
3. Initial-value problems
4. Operators
5. Homogeneous solutions
6. Variation of parameters
7. IVP formula
8. Integrating factor
9. Undetermined coefficients
10. Modeling

III. Steps and impulses

1. Overview
2. Step forcing
3. Impulses
4. Laplace transforms
5. Solutions by transform

IV. Second-order linear ODE

1. Overview
2. Structure of solutions
3. Homogeneous solutions
4. Complex exponentials
5. Pure oscillation
6. Free oscillations
7. Undetermined coefficients
8. Forced oscillations

V. Laplace transforms

1. Overview
2. Solutions to 2nd-order ODEs
3. Steps and impulses (2nd order)
4. Convolution

VI. Linear ODE systems

1. Overview
2. Conversion of higher-order ODEs
3. Solution structure
4. Homogeneous solutions
5. Constant-coefficient problems
6. Stability and the phase plane
7. The matrix exponential
8. Variation of parameters

VII. Nonlinear ODEs

1. Nonlinear ODEs
2. Autonomous scalar problems
3. Separable equations
4. The logistic equation
5. Direction fields
6. Linearization

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By Tobin A. Driscoll
© Copyright 2020.