MATH225 - Differential Equations

# Course description - [**Course Description**](https://catalog.mines.edu/coursesaz/math/): Classical techniques for first and higher order equations and systems of equations. Laplace transforms. Phase-plane and stability analysis of non-linear equations and systems. Applications from physics, mechanics, electrical engineering, and environmental sciences. - **Prerequisites**: Grade of C- or better in MATH112 or MATH122. - **Corequisites**: CSCI128 or CSCI102. 3 hours lecture; 3 semester hours. # Lectures - [[MATH225F26—Live(archived - Vocabulary and theory and ay''+by'+cy=f(t)).pdf]] - [[MATH225F26—Live(archived - Geometry and symbolic solutions).pdf]] - [[MATH225F26(Day 1) - Introduction to the course, and what an ODE is|Day 1 (Mon Aug 24) — Introduction to the course, and what an ODE is]] - [[MATH225F26(Day 2) - Order, type, linearity, and what it means to solve an ODE|Day 2 (Wed Aug 26) — Order, type, linearity, and what it means to solve an ODE]] - [[MATH225F26(Day 3) - Verifying solutions, explicit and implicit, and the existence-uniqueness theorem|Day 3 (Fri Aug 28) — Verifying solutions, explicit and implicit, and the existence-uniqueness theorem]] - [[MATH225F26(Day 4) - Initial value problems, intervals of definition, and when the theorem is silent|Day 4 (Mon Aug 31) — Initial value problems, intervals of definition, and when the theorem is silent]] - [[MATH225F26(Day 5) - Slope fields, and where uniqueness breaks|Day 5 (Wed Sep 2) — Slope fields, and where uniqueness breaks]] - [[MATH225F26(Day 6) - Phase lines, classifying equilibria, and the first solution technique|Day 6 (Fri Sep 4) — Phase lines, classifying equilibria, and the first solution technique]] - [[MATH225F26(Day 7) - A table for the phase line, partial fractions, and which root to keep|Day 7 (Wed Sep 9) — A table for the phase line, partial fractions, and which root to keep]] - [[MATH225F26(Day 8) - The integrating factor method, and the interval that comes with it|Day 8 (Fri Sep 11) — The integrating factor method, and the interval that comes with it]] - [[MATH225F26(Day 9) - Undetermined coefficients, and the method of the lucky guess|Day 9 (Mon Sep 14) — Undetermined coefficients, and the method of the lucky guess]] - [[MATH225F26(Day 10) - The MUC table, resonance, and the Bernoulli substitution|Day 10 (Wed Sep 16) — The MUC table, resonance, and the Bernoulli substitution]] - [[MATH225F26(Day 11) - Finishing Bernoulli, and what a model is|Day 11 (Fri Sep 18) — Finishing Bernoulli, and what a model is]] - [[MATH225F26(Day 12) - Tanks with a changing volume, and Newton's law of warming|Day 12 (Mon Sep 21) — Tanks with a changing volume, and Newton's law of warming]] - [[MATH225F26(Day 13) - The oven's temperature, logistic growth, and what harvesting does|Day 13 (Wed Sep 23) — The oven's temperature, logistic growth, and what harvesting does]] - [[MATH225F26(Day 14) - Two tanks solved one at a time, and the first second-order equation|Day 14 (Fri Sep 25) — Two tanks solved one at a time, and the first second-order equation]] - [[MATH225F26(Day 15) - Linear independence, the Wronskian, and real or repeated roots|Day 15 (Mon Sep 28) — Linear independence, the Wronskian, and real or repeated roots]] - [[MATH225F26(Day 16) - Complex roots, Euler's formula, and initial versus boundary conditions|Day 16 (Wed Sep 30) — Complex roots, Euler's formula, and initial versus boundary conditions]] - [[MATH225F26(Day 17) - Why Euler's formula holds, and undetermined coefficients at second order|Day 17 (Fri Oct 2) — Why Euler's formula holds, and undetermined coefficients at second order]] - [[MATH225F26(Day 18) - A repeated root that needs t squared, and choosing a first-order method|Day 18 (Mon Oct 5) — A repeated root that needs t squared, and choosing a first-order method]] - [[MATH225F26(Day 19) - The mass-spring model, and a last review of first-order methods|Day 19 (Wed Oct 7) — The mass-spring model, and a last review of first-order methods]] - [[MATH225F26(Day 20) - Simple harmonic motion, reading a spring word problem, and three kinds of damping|Day 20 (Wed Oct 14) — Simple harmonic motion, reading a spring word problem, and three kinds of damping]] - [[MATH225F26(Day 21) - The three damped cases, and why sinusoidal forcing is the interesting kind|Day 21 (Fri Oct 16) — The three damped cases, and why sinusoidal forcing is the interesting kind]] - [[MATH225F26(Day 22) - Solving the resonant case, and resonance in the wild|Day 22 (Mon Oct 19) — Solving the resonant case, and resonance in the wild]] - [[MATH225F26(Day 23) - Beats near resonance, and the Laplace transform from its definition|Day 23 (Wed Oct 21) — Beats near resonance, and the Laplace transform from its definition]] - [[MATH225F26(Day 24) - Products don't split, the transform of a derivative, and the first Laplace IVPs|Day 24 (Fri Oct 23) — Products don't split, the transform of a derivative, and the first Laplace IVPs]] - [[MATH225F26(Day 25) - Second-order IVPs by Laplace, with partial fractions and completing the square|Day 25 (Mon Oct 26) — Second-order IVPs by Laplace, with partial fractions and completing the square]] - [[MATH225F26(Day 26) - Two more table rules, the Heaviside switch, and seeing a step-function solution coming|Day 26 (Wed Oct 28) — Two more table rules, the Heaviside switch, and seeing a step-function solution coming]] - [[MATH225F26(Day 27) - Step-function practice, transform then shift and shift then transform|Day 27 (Fri Oct 30) — Step-function practice, transform then shift and shift then transform]] # Video library - [[Teaching/MATH225/05 Video Library/Introduction to Differential Equations|Introduction to Differential Equations]] — the course video library: 159 short lecture videos with written notes, in three Parts. Every lecture note above links to the videos that match it. # Archived material ## Archived videos from the pandemic * [[Teaching/MATH225/Video Resources/(Part 1-Videos) Definitions, terminology, theoretical questions, geometry, solution techniques, and models.|(Part 1-Videos) Definitions, terminology, theoretical questions, geometry, solution techniques, and models.]] * [[Teaching/MATH225/Video Resources/(Part 2-Videos) Theory for second-order linear equations, general solutions for homogeneous equations with constant coefficients, the method of undetermined coefficients, electrical and mechanical oscillatory systems, and the Laplace transform|(Part 2-Videos) Theory for second-order linear equations, general solutions for homogeneous equations with constant coefficients, the method of undetermined coefficients, electrical and mechanical oscillatory systems, and the Laplace transform]] * [[Teaching/MATH225/Video Resources/(Part 3-Videos) Systems of autonomous ordinary differential equations, linear equations, and linearization of nonlinear equations|(Part 3-Videos) Systems of autonomous ordinary differential equations, linear equations, and linearization of nonlinear equations]]