**NOTE:**This webpage contains a tentative course plan, suggested exercises, lecture notes, and exam information for MATH 2552C-C01-C02-C03, Spring 2021. It may be updated regularly during the course. Please visit frequently. -MH-

**Lectures:**TR 9:30-10:45am ET by Marc Härkönen, on BlueJeans synchronously and recorded**Studios Section C01:**MW 12:30-1:20pm ET by Yian Yao, on BlueJeans synchronously and recorded**Studios Section C02:**MW 2:00-2:50 ET by Yian Yao, on BlueJeans synchronously and recorded**Studios Section C03:**MW 2:00-2:50 ET by Sarah Eisenstadt, on BlueJeans synchronously and recorded**Text:**James R. Brannan and William E. Boyce,**Differential Equations**(an introduction to modern methods & applications),**3rd edition**, John Wiley & Sons, 2015. [Publisher's website for the text]

01/14 (Thu) First lecture. Course description. What do we need for Math2552? What do we learn in Math2552? 1.1 Diff Eqs, Math Models. Slides 1.2 Direction Fields. Exercises: 1.2 (14,19,25,26,28) 01/18 (Mon) Official Institute Holiday 01/19 (Tue) 1.3 Classification of Differential Equations (orders, single eqs vs systems, linear vs nonlinear). Slides. Exercises: 1.3 (1-6,13,16,21,23,32) 2.1 Separable equations: dy/dt=f(t)g(y). Slides. Exercises: 2.1 (29,35, and click here) 2.2 Linear diff eqs of 1st order: Method of integrating factor. Slides. Homogeneous linear equations: dy/dt+a(t)y=0 Nonhomogeneous linear equations: dy/dt+a(t)y=b(t) Exercises: 2.2 (click here) 01/20 (Wed) First studio. 01/21 (Thu) 2.3 Modeling with 1st order eqs. Slides. Electric circuits. Exercises: 2.3 (1,3,5,25), 1.1 (4,12,14). Some Fun: If you are a sports maniac, try 2.3 (32,33). 01/26 (Tue) 2.4 Differences between homog linear, nonhomog linear, & nonlinear eqs. Slides. Intervals of existence of solutions. Slides. Exercises: 2.4 (1,3,5, and click here) 01/28 (Thu) 2.5 Autonomous eqs and population dynamics. Slides. Exercises: 1.2 (1,9), 2.5 (1,2) 2.5 Phase Portraits for Autonomous equations. Slides. Stability, asymptotic stability, and instability. Slides. See p.458 for the formal definitions of "stable", "unstable" and "asymptotically stable". This Scholarpedia article may be helpful too. Exercises: Click here 02/01 (Monday) ---------Quiz 1--------- Release: 02/01 (Monday) at the beginning of your studio session on Gradescope. Submission: Scan your solutions into a PDF file and submit to Gradescope by the end of your studio time on 02/01 (Monday). Coverage: Lectures & Studios of 01/14-01/25 Policy: Take home. Construct solutions by yourself. No discussion with other people in any form. Please abide by the Honor Code. 02/02 (Tue) 3.2 Two-dimensional Linear systems of diff eqs Exercises: 3.2 (1-8) 6.1 & 6.2 n-dimensional Linear systems of diff eqs. Exercises: 6.1 (2), 6.2 (8,9) Review Linear Algebra: (Section 3.1 can be of some help) how to solve linear systems Ax=b, determinants, eigenvalues, eigenvectors 02/04 (Thu) 3.3 2-D Homog linear systems with const coefficients (distinct real eigenvalues): Solution formula. Exercises: 3.3 Find general solutions in (5,6,10,11,12). Solve the initial value problem in (15,16). 6.3 n-D Homog linear systems with const coefficients (distinct real eigenvalues): Solution formula. Slides. Exercises: 6.3 (4,10,16,21) 3.3 Phase portraits of 2-d systems: Distinct nonzero eigenvalues. (Example 1: Attractive improper node) Exercises: 3.3 (6,17) 02/09 (Tue) 3.3 Phase portraits of 2-d systems: Distinct nonzero eigenvalues. (Examples 2 and 3: Repulsive improper node. Saddle.) Exercises: 3.3 (10,11,15,16,19,23) 3.3 Phase portraits of 2-d systems: Zero eigenvalues. Exercises: 3.3 (5,12) 3.4 Two-dim Homogeneous linear systems with const coefficients (complex eigenvalues). Slides. Exercises: 3.4 (2,4,5,6,7) 6.4 n-D Homogeneous linear systems with const coefficients (complex eigenvalues). Slides. Exercises: 6.4 (4,6) Supplemental file: A Catalogue of Phase Portraits of Homog 2-D Linear Systems 02/11 (Thu) [continue] 3.4 abd 6.4 Complex eigenvalues 3.5 Homog 2-D linear systems with const coefficients (repeated real eigenvalues). Slides. Exercises: 3.5 (2,3,10) 02/15 (Monday) ---------Quiz 2--------- Release: 02/15 (Monday) at the beginning of your studio session on Gradescope. Submission: Scan your solutions into a PDF file and submit to Gradescope by the end of your studio time on 02/15 (Monday). Coverage: Lectures & Studios of 01/26-02/08 Policy: Take home. Construct solutions by yourself. No discussion with other people in any form. Please abide by the Honor Code. 02/16 (Tue) [continue] 3.5 Repeated real eigenvalues Shifted systems of the form:x'=A(x-a).x'=Ax+b. Slides Exercises: click here 02/18 (Thu) Applications: salt in several tanks Exercises: p.144 (31), p.437 (12)~~Applications: electric circuits~~(skip)~~Exercises: p.144 (29), p.167 (32), p.177 (21)~~4.1 Second order linear eqs: definitions and examples 4.2 Second order linear homogeneous eqs: y''+p(t)y'+q(t)y=0. Slides. Exercises: 4.1 (1-9,16,17,21), and click here 4.3 Second order linear homogeneous eqs with const coefficients: ay''+by'+cy=0. Slides. Distinct nonzero eigenvalues. Exercises: 4.3 (2,11,19,30,43) 02/23 (Tue) 4.3 Second order linear homogeneous eqs with const coefficients: ay''+by'+cy=0. Slides. Complex eigenvalues. Exercises: 4.3 (6,23,25,36,29) Repeated eigenvalues. Exercises: 4.3 (7,21,29) Zero eigenvalues. Exercises: 4.3 (13) 02/25 (Thursday) ---------Midterm 1--------- Release: 02/25 (Thursday) 9:30am Eastern Time, on your studio Gradescopepage. Submission: Scan your solutions into a PDF file and submit to Gradescope by 02/25 (Thursday) 10:45am Eastern Time. Coverage: Lectures & Studios of 01/14-02/17 Policy: Take home. Construct solutions by yourself. No discussion with other people in any form. Please abide by the Honor Code. 03/02 (Tue) 4.4 Free vibrations. Spring-mass systems. slides Exercises: 4.4 (3,5,6,11,17) Series RLC circuits. slides. Exercises: 4.4 (8,12,18) 03/04 (Thu) 4.5 Nonhomogeneous linear eqs (method of undetermined coefficients). (Slides) Supplementary material: Higher order linear eqs with constant coefficients Exercises: 4.5 (1-5,18,21,26,27,37) 03/09 (Tue) 4.5 Nonhomogeneous linear eqs (method of undetermined coefficients). [continued] 4.6 Forced vibrations. Resonance. slides Exercises: 4.6 (9,12,15,16,17) 03/10 (Wednesday) ---------Quiz 3--------- Release: 03/10 (Wednesday) at the beginning of your studio session on Gradescope. Submission: Scan your solutions into a PDF file and submit to Gradescope by the end of your studio time on 03/10 (Wednesday). Coverage: Lectures & Studios of 02/18-03/03 Policy: Take home. Construct solutions by yourself. No discussion with other people in any form. Please abide by the Honor Code. 03/11 (Thu) 4.6 Forced vibrations (continued) 4.7 Nonhomogeneous linear eqs (variation of parameters). (Slides) Exercises: 4.7 (3,7,11,19) 03/16 (Tue) Mid-semester break. No classes, assignments, or assessments. 03/17 (Wednesday) Withdrawal Deadline Last day to withdraw from a single course or from school with "W" grades for Spring Semester 2021 by 4:00 pm Eastern Time. 03/18 (Thu) 5.1 Definition of the Laplace transform. (Slides) 5.2 Properties of the Laplace transform ℒ{c}, ℒ{e^{at}}, ℒ{cos(at)}, ℒ{sin(at)}, ℒ{e^{at}sin(bt)}, ℒ{e^{at}sin(bt)}, ℒ{t}, ℒ{^{n}t}. ℒ{^{n}e^{at}t(^{n}ft)}. ℒ{e^{ct}f(t)}. Exercises: 5.1 (13,15,16,21,23), 5.2 (5,7,9,11a) 5.3 The inverse Laplace transform Exercises: 5.3 (9,10,13,14) * Review partial fractions. 03/23 (Tue) 5.2 Derivative Formulas: ℒ{f'(t)}, ℒ{f^{(k)}(t)}. Exercises: 5.2 (18,19,20,24) * Transform the initial value problem, y'(t)-4y(t)=2-5t, y(0)=7, into an algebraic equation for Y(s)=ℒ{y(t)}. Then find Y(s). 5.3 The inverse Laplace transform Exercises: 5.3 (11,18,19,21,23) 5.4 Solve diff eqs with Laplace transforms. (Slides) Exercises: 5.4 (1,2,5,10,12,18) 03/24 (Wed) Mid-semester break. No classes, assignments, or assessments. 03/25 (Thu) 5.4 Solve diff eqs with Laplace transforms (continued). 5.5 Discontinuous functions (Slides) Exercises: 5.5 (2,8,11,14,15) 03/29 (Monday) ---------Quiz 4--------- Release: 03/29 (Monday) at the beginning of your studio session on Gradescope. Submission: Scan your solutions into a PDF file and submit to Gradescope by the end of your studio time on 03/29 (Monday). Coverage: Lectures & Studios of 03/04-03/22 Policy: Take home. Construct solutions by yourself. No discussion with other people in any form. Please abide by the Honor Code. 03/30 (Tue) 5.5 Periodic functions (Slides) Exercises: 5.5 (22) 5.6 Diff eqs with discontinuous forcing functions (Slides) Exercises: 5.6 (2,4,6,12) 04/01 (Thu) 5.7 Impulse function. Delta function. Exercises: 5.7 (1,2,3,9,10,17,18) 5.8 Convolution. Slides Transfer Function & Impulse Response. Slides Exercises: 5.8 (2,4,6,8,9,12,16,20) 04/06 (Tue) 7.1 Autonomous systems. Stability. (Slides) Exercises: Consider 6.3 (1,2,3,4), 6.4 (2,4,6,13), 6.7 (1,3,5,8). Is the equilibrium (critical point)x=0asymptotically stable, stable, or unstable? Ans: 6.3 (AS, US, US, S but not AS), 6.4 (US, AS, S but not AS, S but not AS), 6.7 (US,US,US,AS) 7.2 Almost linear systems in N-dimensions. (Slides) Exercises: 2D systems, and 3D systems 04/08 (Thursday) ---------Midterm 2--------- Release: 04/08 (Thursday) 9:30am Eastern Time, on your studio Gradescopepage. Submission: Scan your solutions into a PDF file and submit to Gradescope by 04/08 (Thursday) 10:45am Eastern Time. Coverage: Lectures & Studios of 02/18-03/31 Policy: Take home. Construct solutions by yourself. No discussion with other people in any form. Please abide by the Honor Code. 04/13 (Tue) A First Order Nonlinear Diff Eq and its Linear Aprroximating Eq near an equilibrium (Slides) Exercises: Click here 7.3 Competing species. Slides. Exercises: 7.3 (1,2) 04/15 (Thu) 7.4 Predator-prey systems. Slides. Exercises: 7.4 (1,3) 7.6 Chaos in the Lorenz attractor. Slides. "Chaos Theory" on Wikipedia 04/20 (Tue) 8.1 Numerical approximations: Euler's method. (Slides) 8.2 Accuracy of numerical methods Exercises: 8.1 (3,6), 8.2 (7,16) 04/21 (Wednesday) ---------Quiz 5--------- Release: 04/21 (Wednesday) at the beginning of your studio session on Gradescope. Submission: Scan your solutions into a PDF file and submit to Gradescope by the end of your studio time on 04/21 (Wednesday). Coverage: Lectures & Studios of 04/01-04/14 Policy: Take home. Construct solutions by yourself. No discussion with other people in any form. Please abide by the Honor Code. 04/22 (Thu) 8.3 Improved Euler method. The Runge-Kutta method. (Slides) Exercises: 8.3 (7,11) 04/26 (Mon) Last Studio. Review. 04/27 (Tue) Last Lecture. Review. (Slides) May 6 (Thursday) ---------Final Exam--------- Release: May 6 (Thursday) 8:00am Eastern Time, on Gradescope. Submission: Scan your solutions into a PDF file and submit to Gradescope by May 6 (Thursday) 10:50am Eastern Time. Coverage: the whole semester Policy: Take home. Construct solutions by yourself. No discussion with other people in any form. Please abide by the Honor Code.