Math 356: Elementary Differential Equations (Spring 2026)
Instructor: Di Fang
Please find the Course webpage on Canvas! This page is generated from Canvas and intended solely for maintaining a record of the teaching schedule and may not be consistently updated.
Please find the references (textbooks) for the course in the syllabus.
Math 356 is an undergraduate-level differential equations course designed primarily for mathematics majors and students interested in learning more of the mathematical foundations of differential equations. The department also offers Math 353, a more engineering-oriented version of the course.
Prerequisites: Linear algebra (the equivalent of 218 or 221); Multivariable Calculus(the equivalent of 202, 212, or 222
The course will cover:
- Part 1 Fundamentals and first-order scalar differential equations (solution techniques; existence and uniqueness)
- Part 2 Second-order scalar differential equations and two-dimensional linear systems
- Part 3 Laplacian Transform and its application to solve differential equations
- Part 4 General n-dimensional linear system of ODEs (homogeneous and nonhomogeneous)
- Part 5 Nonlinear systems
- Part 6 Partial Differential Equations (bounded domain); Fourier series; separation of variables
- Part 7 Partial Differential Equations (infinite domain); Fourier transform
- Part 8 (optional) Numerical methods of differential equation; Optimization and gradient descent
| Week | Monday | Wednesday | HW |
| 1 | Motivation and terminology Math356_lec1.pdf |
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| 2 | The importance of the interval $I$; How to solve first-order ODEs? Some techniques from Calculus -- Old and New (substitution; IBP; partial fraction decomposition); Separable DEs; Integrating Factor Method. Math356_lec2.pdf |
Techniques continued; linearity; mixing problems. (Mixing problems will appear on HW2) | HW1 (due Jan 23) |
| 3 | Martin Luther King Day (no class) | Need for Theory; statements of well-posedness theorems + understanding (existence, uniqueness; no proofs to both theorems -- we discussed how to use the theorems to prove some other statements). Math356_lec4.pdf |
HW2 (due Jan 30) |
| 4 | Exact Differential Equation; Variation of Constant Method. | Second-order Differential Equation (solution space structure; find general solution in the case of constant coefficients; Wronskian); Revisit some linear algebra; linearity principle for a system of first-order linear ODEs. Math356_lec6.pdf |
HW3 (due Feb 6) |
| 5 |
Finish 2nd-order scalar linear ODE and Wronskian; revisit linear algebra (linear space, span, basis, dimension, etc); System of first-order linear ODEs (solution space = a linear space of dimension n; with proofs) |
Fundamental matrix; Revisit matrix diagonalization and the Jordan canonical form; Matrix exponential; How to compute fundamental matrix (with constant coefficients). Math356_lec8.pdf |
HW4 (due Feb 13) This HW is based on material from Week 5 and is therefore covered in the midterm. |
| 6 | more on fundamental matrix - general A (n by n) with Jordan blocks (non-diagalizable case); generalized eigenvectors; Midterm Q&As Math356_lec9.pdf |
Midterm (The midterm will cover material taught during the first five weeks of the lecture). | HW5 (due Feb 20) |
| 7 | More on fundamental matrix - more on Joran chains and complex eigen-pairs Math356_lec10.pdf |
One last note on Jordan chains; Linearity Principle and Variation of Constant -- general case Math356_lec11.pdf |
HW6 (due Feb 27) |
| 8 | 2D phase portrait: revisit of 2D fundamental matrix; stable/unstable/ asymptotically stable; improper and proper nodes (simple case), center (general case). Math356_lec12.pdf |
2D phase portrait, cont'd; finish all cases. Finished Case 3: complex eigenpair (spiral); Case 1: real eigenvalues + diagonalizable (improper or proper node); Case 2: real eigenvalues with Jordan block. Math356_lec13.pdf |
HW7 (due Mar 6) |
| 9 | trace-determinant plane; nonlinear differential equations (systems), linearization, statement of the Hartman-Grobman theorem. Math356_lec14.pdf (The presentation order and notation—such as the choice of letters used to denote variables—in the notes may not be exactly the same as in the lecture, as I often adjust details during class.) |
Finish nonlinear systems' stability; Laplace transform Math356_lec15.pdfDownload Math356_lec15.pdf |
HW8 (due Mar 20) |
| 10 | Spring Break |
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| 11 | more on Laplace transform; convolution. Math356_lec16.pdfDownload Math356_lec16.pdf |
Fourier series Math356_lec17.pdfDownload Math356_lec17.pdf |
HW9 (due Mar 27) |
| 12 | more on Fourier series -- convergence theorem, complex-version of Fourier series; Solve heat equation with Dirichlet boundary conditions Math356_lec18.pdfDownload Math356_lec18.pdf Fancy animes: Gibbs phenomenonLinks to an external site. draw Fourier using Fourier seriesLinks to an external site. |
Solve heat equation with periodic boundary condition; Parseval's identity for Fourier series; Intro to Fourier Transform. Math356_lec19.pdfDownload Math356_lec19.pdf |
HW10 (due Apr 3) |
| 13 |
More on Fourier Transform, Properties, convolution, Dirac delta function, Use Fourier Transform to solve PDE; heat kernel. |
More on Dirac delta function; Use Fourier transform to solve transport equation, heat equation; Derivation of heat equation from random walk.
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HW11 (due Apr 10) |
| 14 | Duhamel's principle to solve nonhomogeneous linear PDEs; separation of variables; Sturm-Liouville Problem. Math356-lec22.pdfDownload Math356-lec22.pdf |
More on separation of variables + other boundary conditions (Dirichlet, Neumann, Mixed; heat & wave equations); the idea of eigenvalue and eigenfunction. Math356_lec23.pdfDownload Math356_lec23.pdf |
HW12 (due Apr 17) This is the last HW of the course! No more! |
| 15 (optional materials) |
numerical method for ODE, finite difference, explicit versus implicit schemes, numerical analysis (local truncation error & global error analysis) code demo (in jupyter notebook via Python)Download code demo (in jupyter notebook via Python) Math356_lec24.pdfDownload Math356_lec24.pdf |
Numerical method for PDEs; central difference, to solve heat equation/Schrodinger equatoin; Trotter formula (splitting method), discrete Fourier transform (DFT). code demo (in jupyter notebook via Python)Download code demo (in jupyter notebook via Python) optimization from ODE view point, gradient flow, gradient descent, stochastic gradient descent code demo (in jupyter notebook via Python)Download code demo (in jupyter notebook via Python) |
No HW |
| 16 |
Course Review |
Final Q&A | No HW |