MATH 2552K, Spring 2022 (Course Topics, Exercises, Notes, and Exam Info)

Spring 2022, MATH 2552K Differential Equations
Xu-Yan Chen
NOTE: This webpage contains a tentative course plan, lecture slides, suggested exercises, and exam information for MATH 2552K, Spring 2022. It may be updated regularly during the course. Please visit frequently. -XYC-
Lectures Section K: TR 2:00pm-3:15pm ET by Xu-Yan Chen, Howey Physics L2 and synchronously on BlueJeans and recorded
Studios Section K01: MW 3:30pm-4:20pm ET by Ian Lewis, Skiles 168
Studios Section K02: MW 3:30pm-4:20pm ET by Jieun Seong, Skiles 270
Studios Section K03: MW 5:00pm-5:50pm ET by Ian Lewis, Skiles 168
Studios Section K04: MW 5:00pm-5:50pm ET by Jieun Seong, Skiles 270
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]
01/10 (Mon) First Studio
01/11 (Tue) 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)

01/13 (Thu) 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)

01/17 (Mon) Official School Holiday

01/18 (Tue) 2.3 Modeling with 1st order eqs. Slides.
        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).

01/20 (Thu) Electric circuits.
	    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) 
            

01/22 (Sat) WeBWork Deadline: Two HW sets, HW 00 and HW 01, close at 11:59pm ET.

01/24 (Mon) Quiz 1 on CANVAS 
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures 01/11-01/13 & Studios 01/10-01/19
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
	Info: FAQ About Quizzes
              Exam Requirements
              A Quiz Screenshot (with math problems removed)

01/25 (Tue) 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

01/27 (Thu) 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
            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)

01/29 (Sat) WeBWork Deadline: HW 02 closes at 11:59pm ET.

01/31 (Mon) Quiz 2 on CANVAS 
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures 01/18-01/25 & Studios 01/24-01/26
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
	Info: FAQ About Quizzes
              Exam Requirements
              A Quiz Screenshot (with math problems removed)

02/01 (Tue) [continue] 3.3 and 6.3 Homog linear systems with const coefficients (distinct real eigenvalues): Solution formula. 
	    3.3 Phase portraits of 2-d systems: Distinct nonzero eigenvalues. (Examples 1-3: Attractive improper node. Repulsive improper node. Saddle.)
	Exercises: 3.3 (6,10,11,15,16,17,19,23) 

02/03 (Thu) 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

02/07 (Mon) Quiz 3 on CANVAS 
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures & Studios 01/27-02/02
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
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02/08 (Tue) [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)

02/10 (Thu) Shifted systems of the form: x'=A(x-a). x'=Ax+b. Slides
        Exercises: click here
	    Applications: salt in several tanks
        Exercises: p.144 (31), p.437 (12)
	    Applications: electric circuits
        Exercises: p.144 (29), p.167 (32), p.177 (21)

02/12 (Sat) WeBWork Deadline: HW 03 closes at 11:59pm ET.	

02/14 (Mon) Quiz 4 on CANVAS 
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures & Studios 02/03-02/09
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
	Info: FAQ About Quizzes
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              A Quiz Screenshot (with math problems removed)
    
02/15 (Tue) 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)

02/17 (Thu) 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)

02/19 (Sat) WeBWork Deadline: Two HW sets, HW 04 and HW 05, close at 11:59pm ET.

02/22 (Tuesday) ---------Midterm 1 on CANVAS---------
	Start anytime between 2:00pm-3:15pm ET, and submit a five-page PDF file of your work on CANVAS within 60 min and at latest before 3:15pm ET.
        Coverage: Lectures & Studios of 01/10-02/14
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
	Info: FAQ About Midterms
              Exam Requirements
              A Midterm Screenshot (with math problems removed)

02/24 (Thu) 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)

02/26 (Sat) WeBWork Deadline: HW 06 closes at 11:59pm ET.

02/28 (Mon) Quiz 5 on CANVAS 
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures & Studios 02/15-02/23
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
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03/01 (Tue) 4.5 Nonhomogeneous linear eqs (method of undetermined coefficients). (Slides. pages 1-58)
        Exercises: 4.5 (1,2,5,18,37)

03/03 (Thu) 4.5 Nonhomogeneous linear eqs (method of undetermined coefficients). The case of resonance. (Slides. pages 59-98)
	    Supplementary material: Higher order linear eqs with constant coefficients
        Exercises: 4.5 (3,4,21,26,27)
            4.6 Forced vibrations. Resonance. slides
        Exercises: 4.6 (9,12,15,16,17)

03/05 (Sat) WeBWork Deadline: HW 07 closes at 11:59pm ET.

03/07 (Mon) Quiz 6 on CANVAS 
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures & Studios 02/24-03/02
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
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03/08 (Tue) 4.6 Forced vibrations (continued)
	    4.7 Nonhomogeneous linear eqs (variation of parameters). (Slides)
        Exercises: 4.7 (3,7,11,19)

03/10 (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^atsin(bt)}, ℒ{e^at sin(bt)}, ℒ{tⁿ},  ℒ{tⁿe^at}.
            ℒ{tⁿf(t)}.
	    ℒ{e^ctf(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.

03/12 (Sat) WeBWork Deadline: HW 08 closes at 11:59pm ET.

03/14 (Mon) Quiz 7 on CANVAS 
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures & Studios 03/03-03/09
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
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03/15 (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. 1st order examples. (Slides pages 1-4)
        Exercises: 5.4 (click here

03/17 (Thu) 5.4 Solve diff eqs with Laplace transforms. (Slides pages 5-17).
        Exercises: 5.4 (1,2,5,10,12,18)
	    5.5 Discontinuous functions (Slides)
        Exercises: 5.5 (2,8,11,14,15)	

03/18 (Fri) 4:00pm ET, Withdrawal Deadline

03/19 (Sat) WeBWork Deadline: Two HW sets, HW 09 and HW 10, close at 11:59pm ET.

03/21 (Mon) - 03/25 (Fri) Spring Break. No classes, assignments, or assessments.
  
03/29 (Tue) 5.5 Periodic functions (Slides)
        Exercises: 5.5 (22) (skipped)
	    5.6 Diff eqs with discontinuous forcing functions (Slides)
        Exercises: 5.6 (2,4,6,12)
	    
03/31 (Thu) 5.7 Impulse function. Delta function. (Slides)
        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)

04/02 (Sat) WeBWork Deadline: HW 11 closes at 11:59pm ET.

04/05 (Tuesday)---------Midterm 2 on CANVAS---------
	Start anytime between 2:00pm-3:15pm ET, and submit a five-page PDF file of your work on CANVAS within 60 min and at latest before 3:15pm ET.
        Coverage: Lectures & Studios of 02/15-03/28
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
	Info: FAQ About Midterms
              Exam Requirements
              A Midterm Screenshot (with math problems removed)

04/07 (Thu) 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=0 asymptotically 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

04/09 (Sat) WeBWork Deadline: HW 12 closes at 11:59pm ET.

04/11 (Mon) Quiz 8 on CANVAS 
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures & Studios 03/29-04/06
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
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04/12 (Tue) 7.2 Almost linear systems in N-dimensions. [continued]
	    7.3 Competing species. Slides.
        Exercises: 7.3 (1,2)

04/14 (Thu) A First Order Nonlinear Diff Eq and its Linear Aprroximating Eq near an equilibrium (Slides) 
	Exercises: Click here
	    7.4 Predator-prey systems. Slides.
        Exercises: 7.4 (1,3)
	    7.6 Chaos in the Lorenz attractor. Slides.
            "Chaos Theory" on Wikipedia

04/16 (Sat) WeBWork Deadline: HW 13 closes at 11:59pm ET.

04/18 (Mon) Quiz 9 on CANVAS
	Start anytime between 9am-10pm ET, and submit a two-page PDF file of your work on CANVAS within 30 min and at latest before 10pm ET.
	Coverage: Lectures & Studios 04/07-04/13
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
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04/19 (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 (Thu) 8.3 Improved Euler method. The Runge-Kutta method. (Slides)
        Exercises: 8.3 (7,11)

04/23 (Sat) WeBWork Deadline: Two HW sets, HW 14 and HW 15, close at 11:59pm ET.

04/25 (Mon) Last Studio. Review.
04/26 (Tue) Last Lecture. Review. (Slides)

April 28 (Thursday) ---------Final Exam on CANVAS---------
	Start anytime between 2:40pm-5:30pm ET, and submit a PDF file of your work on CANVAS within 2 hr 30 min and at latest before 5:30pm ET.
        Coverage: the whole semester
        Policy: Construct solutions by yourself. No discussion with other people in any form.
 	        Please abide by the Honor Code.
	Info: FAQ About the Final Exam
              Exam Requirements
              A Final Exam Screenshot (with math problems removed)