Notes for Math 351 @ UD

Author

Toby Driscoll

Published

August 12, 2023

Contents

Schedule for Fall 2023

First-order ODEs

Date Section Zill Suggested practice
Aug 30 Intro –
Aug 30 IVPs –
Sep 1 Qualitative methods 2.1 21, 24
Sep 6 Separable equations 2.2 5, 6, 7, 10, 2
Sep 6 Linear equations 2.3 3, 6, 7, 11, 25, 29
Sep 8 Homogeneous problems (2.3)
Sep 8 Variation of parameters (2.3)
Sep 11 Modeling 2.7 1, 3, 5, 15, 23, 27

Second-order ODEs

Date Section Zill Suggested practice
Sep 13 Intro, 2nd order linear problems 3.1 16, 17, 21
Sep 15 Homogeneous solutions 3.3 1-14, 29, 33, 49, 51
Sep 15 Complex solutions 3.3
Sep 18 Variation of parameters 3.5 3, 5, 10, 16
Sep 22 Undetermined coefficients 3.4 1, 3, 5, 8, 31
Sep 20 Oscillators 3.8 1, 6, 27, 31, 33, 43, 44

Linear systems

Date Section Poole Suggested practice
Sep 25 review
Sep 27 Intro to linear systems 2.1 1-5, 19, 20, 21, 27-30, 31
Sep 29 EXAM 1 Goals A1–A13, B1–B9
Oct 2 Row elimination 2.1, 2.2 1-8, 9-14, 25-28, 35-38, 40-41
Oct 4 RRE form 2.2
Oct 6 Linear combinations 2.3 1-4, 7, 9, 11, 22-27

Matrix algebra

Date Section Poole Suggested practice
Oct 9 Elementwise operations 3.1 1, 3, 5, 7, 11, 21-22, 40
Oct 9 Matrix-vector multiplication 3.1
Oct 11 Matrix multiplication 3.1, 3.2 23, 28
Oct 13 Identity and inverse 3.3 1-4 (any method), 13a, 42a, 43a
Oct 13 Fundamental theorem 3.3
Oct 16 Subspaces 3.5 (no row spaces) 11-12, 17-20, 27-28, 35-38, 43, 45-46
Oct 18 Vector spaces 6.1 1, 2, 9, 28, 31, 35, 36, 51-52, 53-54
Oct 20 Coordinates 6.1, 6.2 1-3, 5-7, 18-19, 22-24, 28-29, 51-52
Oct 23 Change of basis 6.3 1-3, 5-7
Oct 25 Review
Oct 27 EXAM 2 Goals C1-C6, D1–D14

Eigenvalues

Date Section Poole Suggested practice
Oct 30 Determinants 4.2 7-12, 45, 47, 49, 50, 57-58
Nov 1 Eigenvalues (properties) 4.1 1-4, 11-12, 27-28, 36
Nov 3 Eigenvalues (computing) 4.3 1-3, 10, 23
Nov 6 Similarity 4.4 1–2,
Nov 6 Diagonalization 4.4 5-6, 8-10, 17–19

Linear ODE systems

Date Section Zill Suggested practice
Nov 8 ODE systems (10.1)
Nov 10 Linear ODE systems 10.1 1-6, 7, 11-12, 17-19
Nov 13 General solutions 10.1 1-6, 7, 11-12, 17-19
Nov 13 Homogeneous equations 10.1 1-6, 7, 11-12, 17-19
Nov 15 Constant-coefficient problems 10.2 1, 5-6, 10, 31, 35-37
Nov 17 Constant-coefficient problems 10.2 1, 5-6, 10, 31, 35-37
Nov 27 Phase plane 11.2 1-5 (part a), 9-16, 19
Nov 29 Review
Dec 1 EXAM 3 Goals E1–E9, F1–F4
Dec 4-6 Matrix exponential 10.5 2, 6, 19-21, 25
Dec 6-8 Linearization 11.3
Dec 11 Review
Dec 13 FINAL EXAM All goals

List of examples

Description link chapter
1st-order ODE standard form Example 1.1 1
Verify ODE solution Example 1.2 1
Variable-growth ODE solution Example 1.3 1
Nonlinear growth ODE solution Example 1.4 1
Solve 1st-order IVP Example 1.5 1
Solution curves and an initial condition Example 1.6 1
Numerical solution for variable growth rate Example 1.8 1
Numerical solution for nonlinear growth Example 1.9 1
Direction field for nonautonomous equation Example 1.11 1
Phase diagram for stability of equilibria Example 1.12 1
Separable equation Example 1.13 1
Separable equation—implicit solution Example 1.14 1
Separable equation—factorization Example 1.15 1
Separable equation—autonomous Example 1.16 1
Linear operator Example 1.17 1
1st order homogeneous solution Example 1.18 1
1st order homogeneous solution Example 1.19 1
1st order homogeneous solution Example 1.20 1
1st order variation of parameters Example 1.21 1
1st order variation of parameters Example 1.22 1
1st order variation of parameters Example 1.23 1
Bacterial growth model Example 1.24 1
Radioactive decay model Example 1.25 1
Cooling coffee model Example 1.26 1
Cooling coffee model Example 1.27 1
Dependent functions Example 2.1 2
Independent functions Example 2.2 2
Wronskian Example 2.3 2
Characteristic equation—double root Example 2.4 2
Characteristic equation—distinct real roots Example 2.5 2
2nd order IVP Example 2.6 2
Complex number fractions Example 2.7 2
Complex number reciprocal Example 2.8 2
Complex numbers polar form Example 2.9 2
Natural frequency of an oscillator Example 2.23 2
Damping strength Example 2.24 2
Free oscillations Example 2.25 2
Two linear equations Example 3.1 3
Coefficient matrix Example 3.2 3
Row elimination Example 3.3 3
Augmented matrix Example 3.4 3
RE form Example 3.5 3
Inconsistent system Example 3.6 3
Infinitely many solutions Example 3.7 3
Gauss–Jordan elimination Example 3.8 3
RE form to RRE form Example 3.9 3
Solution from RRE form Example 3.10 3
RRE form—inconsistent system Example 3.11 3
Linear combination as a linear system Example 3.12 3
Linear dependence Example 3.13 3
Zero vector implies dependence Example 3.14 3
Zero vector again Example 3.15 3
Determine independence of vectors Example 3.17 3
Determine independence of vectors Example 3.17 3
Too many vectors implies dependence Example 3.18 3
Matrix–vector product Example 4.1 4
Matrix–matrix product Example 4.2 4
AB but not BA Example 4.3 4
Matrix multiplication is not commutative Example 4.4 4
Singular matrices Example 4.5 4
Identity matrix Example 4.6 4
Matrix inverse Example 4.7 4
Singular matrix Example 4.8 4
Planes and subspaces Example 4.9 4
The trivial subspace Example 4.10 4
Is it in the column space? Example 4.11 4
Expressing nullspace via span Example 4.12 3
Bases for matrix subspaces Example 4.13 4
Basis for a span Example 4.14 4
Too few to span Example 4.17 4
Too many to be independent Example 4.17 4
Dimension of a subspace Example 4.17 4
Matrices as vectors Example 4.18 4
Polynomials as vectors Example 4.19 4
Coordinates Example 4.20 4
Coordinates relative to a basis Example 4.21 4
Change of basis matrix Example 4.22 4
Change of basis Example 4.23 4
\(3\times 3\) determinant Example 5.1 5
Cramer’s Rule Example 5.2 5
Eigenvectors Example 5.3 5
Imaginary eigenvalues Example 5.4 5
Eigenvectors of the identity matrix Example 5.5 5
Eigenspaces of a triangular matrix Example 5.8 5
Eigenspaces of a \(2\times 2\) Example 5.7 5
Complex eigenvalues Example 5.8 5
Complex eigenvalues Example 5.9 5
Similar matrices Example 5.10 5
Matrix powers Example 5.11 5
Diagonalization Example 5.12 5
Diagonalization shows all Example 5.13 5
Defective eigenvalue Example 5.14 5
Nondefective repeated eigenvalue Example 5.15 5
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Learning goals

A. First-order scalar ODEs

  1. Verify an ODE or IVP solution.
  2. Interpret a direction field for a scalar nonautonomous problem.
  3. Sketch or interpret a phase diagram for an autonomous equation.
  4. Find the steady states/equilibria of an autonomous equation.
  5. Classify the stability of an equilibrium solution.
  6. Solve an initial-value problem if given, or after finding, a general solution.
  7. Solve separable equations that have tractable integrals, in explicit or implicit form.
  8. Distinguish between linear and nonlinear ODEs.
  9. Convert between standard and operator expressions of a linear equation.
  10. Assemble a general solution from homogeneous and particular solutions.
  11. Find the general homogeneous solution of a linear equation.
  12. Use variation of parameters (or an integrating factor) to find a particular solution of a linear equation.
  13. Interpret and apply models of Newtonian cooling/heating, population dynamics, and free fall.

B. Second-order linear ODEs

  1. Assemble a general solution from homogeneous and particular solutions.
  2. Use initial values to solve for the integration constants in a general solution.
  3. Compute the Wronskian of two functions.
  4. Use a Wronskian to determine the linear independence of given solutions.
  5. Solve a homogeneous, linear, constant-coefficient equation by way of its characteristic polynomial.
  6. Use variation of parameters to find a particular solution.
  7. Use undetermined coefficients to find a particular solution with polynomial, exponential, or harmonic forcing.
  8. Relate a linear constant-coefficient equation to a mechanical oscillator, and classify as undamped, underdamped, critically damped, or overdamped.
  9. Identify resonance or pseudoresonance for a harmonically forced oscillator.

C. Linear algebraic systems

  1. Convert between linear systems and matrix–vector representations of them.
  2. Distinguish matrices in RRE form from other matrices.
  3. Perform row elimination to put a matrix into RE or RRE form.
  4. Use the RE or RRE form of a linear system to find its solution(s).
  5. Determine whether a vector lies in the span of other vectors.
  6. Determine whether a set of vectors is linearly independent.

D. Matrix algebra

  1. Convert between a linear combination and an equivalent matrix–vector multiplication.
  2. Compute products of matrices and vectors.
  3. Apply properties (e. g., association, distribution) of matrix algebra.
  4. Recognize or produce an identity matrix.
  5. Apply properties of inverse matrices.
  6. Apply the equivalencies in the Fundamental Theorem of Linear Algebra.
  7. Use a matrix inverse to solve a linear system.
  8. Compute the inverse of a 2x2 or diagonal matrix.
  9. Find a basis for the column space and null space of a matrix.
  10. Find the rank and nullity of a matrix.
  11. Determine whether vectors are a basis of a subspace.
  12. Relate the abstract vector spaces \(\mathbb{R}^{m\times n}\) and \(\mathcal{P}_n\) to ordinary vector spaces.
  13. Compute the coordinates of a given vector in a given basis.
  14. Compute a change-of-basis matrix between two given bases.

E. Eigenvalues

  1. Compute the determinant of a small matrix.
  2. Use a determinant to determine whether a matrix is singular.
  3. Apply the definition and properties of eigenvalues and eigenvectors.
  4. Find eigenvalues and eigenspaces of triangular matrices.
  5. Compute eigenvalues and eigenspaces of 2x2 matrices.
  6. Find the algebraic and geometric multiplicities of an eigenvalue.
  7. Determine whether an eigenvalue or matrix is defective.
  8. Use a similarity transformation to find a matrix power.
  9. Express a nondefective matrix in terms of its diagonalization.

F. Linear ODE systems

  1. Write a linear ODE system in matrix-vector form.
  2. Describe the properties of a fundamental matrix.
  3. Compute a Wronskian and use it to determine the independence of solutions.
  4. Solve a constant-coefficient homogeneous system by eigenvalues and eigenvectors (nondefective case).
  5. Deduce the stability and type of the origin in a homogeneous linear system from its eigenvalues or a phase plane diagram.
  6. Compute a 2x2 matrix exponential.
  7. Use a matrix exponential to solve an IVP.