Summer A 2021 MATH 2030 Section 1: Ordinary Differential Equations
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Basic info
- Lectures: Monday - Thursday 10:45am -12:20pm
- Instructor: Sam DeHority (samdehority@math.columbia.edu)
- Office hours: Monday and Wednesday 2pm-3pm or by appointment
- TA: Andrew Sullivan (ags2198@columbia.edu)
Syllabus
The course syllabus has additional information about the course and a tentative calendar for the whole course.
Calendar
Listed below are the lecture dates, a list of key topics covered during the lecture, and any interactive demonstrations that may be useful.
A note about the interactive demonstrations, which are SageMath Jupyter notebooks: the CoCalc links will be interactive. I will also upload the same notebooks to a GitHub page. If you want to use the GitHub links to the files you should have another way of running SageMath, either by installing it locally on your system, following the directions in the SageMath installation guide or by running it online in a CoCalc notebook.
| Lecture notes | Topics covered | Interactive demo |
| May 3, 2021 | Solution by integration Slope field plotting Integral curves |
Slope Fields and Integral Curves |
| May 4, 2021 | ODE Taxonomy Separation of variables Implicit solutions and IVPs |
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| May 5, 2021 | Existence and uniqueness theorems Vector fields for autonomous systems Stability of equilibria |
Vector fields |
| May 6, 2021 | Solution of 1st order linear ODE Integrating factors Conservative and solenoidal vector fields Exact equations |
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| May 10, 2021 | 2nd order homogeneous equations Characteristic polynomial Complex exponential |
2nd order linear homogeneous equations |
| May 11, 2021 | Repeated roots Wronskian Higher order homogeneous equations Undetermined coefficients |
Resonance |
| May 12, 2021 | Fundamental linear algebra techniques: i) Matrix multiplication ii) Determinants iii) Gaussian elimination iv) Finding eigenvalues v) Finding eigenvectors |
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| May 13, 2021 | Eigenvectors and eigenvalues Homogeneous linear systems with constant coefficients Linear change of coordinates Decoupling |
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| May 17, 2021 | Complex and repeated roots Reducing to 1st order systems Generalized eigenvalues |
Small oscilations |
| May 18, 2021 | Complete solution to $\mathbf{x}’ = A \mathbf{x}$ Inhomogeneous systems Variation of parameters Matrix exponential Jordan blocks |
3d homogeneous constant coefficients systems |
| May 19, 2021 | Change of basis matrix Phase plane Classification of critical points |
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| May 20, 2021 | Midterm review | |
| May 24, 2021 | Qualitative behavior theorems Linearization near critical points Jacobian matrix Power series |
Critical points of the Rössler attractor |
| May 25, 2021 | Power series solutions Recurrence relations Ordinary points |
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| May 26, 2021 | Regular singular points Frobenius method Hypergeometric series |
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| May 27, 2021 | Frobenius method for $r_2 - r_1 \in \mathbb{Z}$ Irregular singular points Möbius transformation Bessel functions |
Hypergeometric and Bessel functions |
| June 1, 2021 | Asymptotics of singular ODEs $\frac{d}{dx}$ in $(x-x_0)^n/n!$ basis Laplace transform |
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| June 2, 2021 | Laplace transform and differentiation Laplace method for IVPs |
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| June 3, 2021 | Discontinuous forcing functions in IVPs step function $u_c(t)$ Distributions Dirac delta $\delta(t-c)$ |
Discontinuous forcing functions |
| June 7, 2021 | Impulse response Convolution |
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| June 8, 2021 | Integral kernels Green’s function |
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| June 9, 2021 | Boundary value problems Neumann and Dirichlet boundary conditions Intro to Sturm-Liouville theory |
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| June 10, 2021 | Final review |
Homeworks
Homeworks are due Monday and Thursday at 11:59pm, to be submitted on Courseworks.
| Due date | File | Solutions |
| May 6 | Homework 1 | |
| May 10 | Homework 2 | |
| May 13 | Homework 3 | |
| May 17 | Homework 4 | |
| May 24 | Homework 5 | |
| May 31 | Homework 6 | |
| June 7 | Homework 7 | |
| June 14 | Homework 8 |