MATH-2200-LN01

Course administration

• Canvas page (where you must log in).
• Help with DjVu (if you have trouble reading the DjVu files on this page).
• Official syllabus (DjVu).
• Course policies (DjVu).
• Class hours: Tuesdays and Thursdays from 4:00 PM to 5:20 in room U6.
• Final exam: May 13 Tuesday from 4:00 PM to 5:40 in room U6 (or by appointment).

Contact information

• Name: Toby Bartels, PhD.
• Canvas messages.
• Email: TBartels@Southeast.edu.
• Voice mail: 1-402-323-3452.
• Text messages: 1-402-805-3021.
• Office hours:
  • on Mondays, Wednesdays, and Fridays from 1:00 PM to 2:00,
  • on Tuesdays and Thursdays from 2:30 PM to 3:30, and
  • by appointment,
  in room U9C and over Zoom meeting 610-024-2876.

Readings

Try to read this introduction before the first day of class:

• Objective: Understand what to expect from this course.
• Reading from my notes: The course policy document (DjVu, same as above).
• Exercises from the textbook due on January 23 Thursday or ASAP thereafter (submit these through MyLab in the Next item): O.1.1, O.1.2, O.1.3, O.1.4, O.1.5, O.1.6, O.1.7, O.1.8, O.1.10, O.1.11, O.1.12.

Basics

1. Introduction:
   • Reading from the textbook: Section 1.1.
   • Exercises due on January 23 Thursday (submit these here on Canvas or in class): Fill in the blanks with vocabulary words:
     1. An equation with derivatives in it is called a(n) _____ equation.
     2. The highest derivative that appears in such an equation is called the _____ of the equation.
     3. A differential equation together with an initial condition is called a(n) _____ (this is a long one).
   • Exercises from the textbook due on January 28 Tuesday (submit these through MyLab in the Next item): 1.1.1, 1.1.3, 1.1.13, 1.1.15, 1.1.17, 1.1.19.
2. Initial examples:
   • Reading from the textbook: Section 1.2, Section 1.4.
   • Exercises due on January 28 Tuesday (submit these here on Canvas or in class): Fill in the blanks with adjectives:
     1. The general solution of dy/dx = f(x) gives y as the _____ integral of f(x) with respect to x.
     2. A differential equation of the form dy/dx = g(x) k(y) is called _____.
     3. If dx/dt = kx for some positive constant k, then x is undergoing _____ growth (relative to t).
   • Exercises from the textbook due on January 30 Thursday (submit these through MyLab in the Next item): 1.2.1, 1.2.4, 1.2.11, 1.2.15, 1.2.17, 1.2.25, 1.2.29, 1.4.1, 1.4.4, 1.4.9, 1.4.13, 1.4.17, 1.4.19, 1.4.22, 1.4.25, 1.4.29, 1.4.33, 1.4.35.
3. Solution curves:
   • Reading from the textbook: Section 1.3.
   • Exercises due on January 30 Thursday (submit these here on Canvas or in class): Consider a differential equation of the form dy/dx = f(x, y).
     1. In forming a slope field for this differential equation, the slope through the point (a, b) on the graph should be _____. (Give a formula involving the data in the question, such as x, y, f, a, and/or b.)
     2. To guarantee that there is a unique solution with the initial value that y = b when x = a, at least for x in some in interval around a, which of the following should be continuous for (x, y) in some rectangle around (a, b)? (Say Yes or No to each.)
        1. f(x, y);
        2. ∂f(x, y)/∂x;
        3. ∂f(x, y)/∂y.
   • Exercises from the textbook due on February 4 Tuesday (submit these through MyLab in the Next item): 1.3.2, 1.3.3, 1.3.5, 1.3.7, 1.3.11, 1.3.13, 1.3.18, 1.3.22, 1.3.27.
4. Numerical solutions:
   • Reading from the textbook: Section 2.4, Section 2.5.
   • Exercises due on February 4 Tuesday (submit these here on Canvas or in class): Consider the initial-value problem dy/dx = f(x, y), with y = y₀ when x = x₀.
     1. If you approximate the solution to this problem using Euler's method with step size h, so that x_n = x₀ + nh, what is the recursive formula for the corresponding approximate solutions y_n? (Use only f, h, values of x, and/or previous values of y.)
     2. If you approximate the solution to this problem using the improved Euler's method with step size h, what are the recursive formulas for the predictors u_n and the the corresponding approximate solutions y_n? (Use only f, h, values of x, and/or previous values of u and/or y.)
   • Exercises from the textbook due on February 6 Thursday (submit these through MyLab in the Next item): 2.4.1, 2.4.5, 2.4.7, 2.5.1, 2.5.5, 2.5.7, 2.5.9.
5. Exact and linear first-order equations:
   • Reading from the textbook: Section 1.6 "Exact Differential Equations" (3 pages), Section 1.5 through "A Closer Look at the Method".
   • Exercises due on February 6 Thursday (submit these here on Canvas or in class):
     1. Given a differential equation M(x, y) + N(x, y) dy/dx = 0, what relationship has to hold between the partial derivatives of M and N, for this to be an exact equation?
     2. Given a differential equation dy/dx + P(x) y = Q(x), what integrating factor should we multiply this by in order to solve it? (This will make it into an exact equation.)
   • Exercises from the textbook due on February 11 Tuesday (submit these through MyLab in the Next item): 1.6.31, 1.6.35, 1.6.37, 1.5.6, 1.5.15, 1.5.24, 1.5.27.
6. Substitution methods:
   • Reading from the textbook: the rest of Section 1.6.
   • Exercises due on February 11 Tuesday (submit these here on Canvas or in class):
     1. Given a differential equation P(x, y) dy/dx = Q(x, y), where P and Q are homogeneous functions with the same degree, what substitution will transform this into a separable equation?
     2. Given a differential equation dy/dx + P(x) y = Q(x) yⁿ, what substitution will transform this into a linear equation?
     3. Given a second-order differential equation for y as a function of x, if y does not appear undifferentiated in the equation, then what substitution will transform this into a first-order equation?
   • Exercises from the textbook due on February 13 Thursday (submit these through MyLab in the Next item): 1.6.1, 1.6.3, 1.6.12, 1.6.19, 1.6.23, 1.6.27, 1.6.46, 1.6.47, 1.6.59.

Linear equations

7. Second-order linear equations:
   • Reading from the textbook: Section 3.1.
   • Exercises due on February 13 Thursday (submit these here on Canvas or in class):
     1. Consider the differential equation D_x²y + p(x) D_xy + q(x) y = 0 (where D_x is d/dx, that is differentiation with respect to x), and let I be an interval on which p and q are both continuous. Suppose that y = y₁(x) and y = y₂(x) are both solutions of this equation on (at least) I, and neither y₁ nor y₂ is a constant multiple of the other. What is the general solution of this differential equation on I?
     2. If y₁ and y₂ are two differentiable functions on some domain, then what is their Wronskian? (Express this using the two functions and/or their derivatives.)
     3. Suppose that λ = λ₁ and λ = λ₂ are distinct real solutions of the equation aλ² + bλ + c = 0. What is the general solution of the differential equation aD_x²y + bD_xy + c = 0?
   • Exercises from the textbook due on February 20 Thursday (submit these through MyLab in the Next item): 3.1.23, 3.1.24, 3.1.1, 3.1.3, 3.1.9, 3.1.13, 3.1.33, 3.1.35, 3.1.37, 3.1.39.
8. Homogeneous-linear equations:
   • Reading from the textbook: Section 3.2 through "General Solutions", Section 3.3.
   • Exercise due on February 20 Thursday (submit this here on Canvas or in class): Suppose that you have a homogeneous-linear ODE with constant real coefficients, for y as a function of x. If the complex roots with multiplicity of the ODE's characteristic polynomial are 0, 0, 1, 2, 2, 2, 3i, 3i, −3i, −3i, 4 + 5i, and 4 − 5i, then what is its general solution? (Write this as a real-number function.)
   • Exercises from the textbook due on February 25 Tuesday (submit these through MyLab in the Next item): 3.2.1, 3.2.7, 3.2.13, 3.2.27, 3.2.33, 3.3.1, 3.3.8, 3.3.11, 3.3.13, 3.3.21, 3.3.23, 3.3.25, 3.3.31.
9. Non-homogeneous equations:
   • Reading from the textbook: Section 3.2 "Nonhomogeneous Equations", Section 3.5 through "The Case of Duplication" (3 pages).
   • Exercise due on February 25 Tuesday (submit this here on Canvas or in class): Suppose that you have a differential equation of the form Ly = f(x), where L is a homogeneous-linear differential operator in x of degree n. If the general solution of the equation Ly = 0 is y = y_c (where y_c is a function of x and n arbitrary constants), and y = y_p (where y_p is a function of x) is one particular solution of Ly = f(x), then what is the general solution of Ly = f(x)?
   • Exercises from the textbook due on February 27 Thursday (submit these through MyLab in the Next item): 3.2.23, 3.5.1, 3.5.2, 3.5.3, 3.5.4, 3.5.19, 3.5.31, 3.5.35, 3.5.37.
10. Harmonic motion:
    • Reading from the textbook: Section 3.4, Section 3.6, Section 3.7.
    • Exercises due on February 27 Thursday (submit these here on Canvas or in class): Consider the differential equation D_t²x + 2pD_tx + ω₀²x = F(t), where p ≥ 0 and ω₀ > 0 are constants and F is a continuous function, describing the behaviour over time (measured by t) of a physical system described by the quantity x. Fill in the following blanks with equations or inequalities involving p, ω₀, and/or F(t):
      1. The system is free if _____, and forced if _____.
      2. The system is undamped if _____, underdamped if _____, critically damped if _____, and overdamped if _____.
      3. Suppose that p = 0 and F has the form F(t) = F₀ cos (ωt), where F₀ ≠ 0 and ω > 0 are constants (so this answer might also refer to F₀ and/or ω, but not p). The system is in pure resonance if _____.
    • Exercises from the textbook due on March 4 Tuesday (submit these through MyLab in the Next item): 3.4.3, 3.4.13, 3.4.15, 3.6.1, 3.6.7, 3.6.17, 3.7.3, 3.7.9, 3.7.17, 3.7.13.
11. Tricky examples:
    • Reading from the textbook: Section 3.5 "Variation of parameters", Section 3.8.
    • Exercises due on March 4 Tuesday (submit these here on Canvas or in class):
      1. Suppose that you have a monic second-order linear ODE, so D_x²y + p(x) D_xy + q(x) y = f(x). Suppose that you know the solutions to the complementary homogeneous-linear equation (with 0 on the right-hand side instead of f(x)), so y_c = c₁ y₁(x) + c₂ y₂(x). Express the general solution of the original equation using integrals involving the functions f, y₁, and/or y₂.
      2. Recall that the initial-value problem D_x²y = F(x, y, D_xy), with y = y₀ and D_xy = y₁ when x = x₀, always has a unique solution for y as a function of x, at least for x in some interval, as long as F is sufficiently differentiable. Must the boundary-value problem D_x²y = F(x, y, D_xy), with y = y₀ when x = x₀ and y = y₁ when x = x₁, also have a unique solution for y as a function of x, at least for x in some interval, as long as F is sufficiently differentiable?
    • Exercises from the textbook due on March 6 Thursday (submit these through MyLab in the Next item): 3.5.53, 3.5.54, 3.8.3, 3.8.4.

Systems of equations and applications

12. Systems of differential equations:
    • Reading from the textbook: Section 4.1.
    • Exercises due on March 6 Thursday (submit these here on Canvas or in class): Suppose you have a system of 3 ordinary differential equations in the 3 dependent variables x, y, and z (in addition to the independent variable t), of order 2 in x, order 3 in y, and order 4 in z, and you convert this into a system of first-order equations.
      1. How many equations will be in the system of first-order equations?
      2. How many dependent variables will this new system have? (Don't count the independent variable t.)
      3. If you can solve this system algebraically for the derivatives and all of the fine print about differentiability is met, then how many constants would you expect in the general solution?
    • Exercises from the textbook due on March 18 Tuesday (submit these through MyLab in the Next item): 4.1.1, 4.1.3, 4.1.5, 4.1.7, 4.1.13.
13. Solving systems by elimination:
    • Reading from the textbook: Section 4.2.
    • Exercises due on March 18 Tuesday (submit these here on Canvas or in class):
      1. If L₁ and L₂ are differential operators of the form a_n(t) D_tⁿ + a_n−1(t) D_tⁿ⁻¹ + ⋯ + a₂(t) D_t² + a₁(t) D_t + a₀(t), then is it always true that L₁L₂ = L₂L₁?
      2. If L₁ and L₂ are differential operators of the form a_nD_tⁿ + a_n−1D_tⁿ⁻¹ + ⋯ + a₂D_t² + a₁D_t + a₀, then is it always true that L₁L₂ = L₂L₁?
    • Exercises from the textbook due on March 25 Tuesday (submit these through MyLab in the Next item): 4.2.3, 4.2.5, 4.2.9, 4.2.11, 4.2.13, 4.2.15, 4.2.19, 4.2.32.
14. Population models:
    • Reading from the textbook: Sections 2.1 and 2.2.
    • Exercises due on March 25 Tuesday (submit these here on Canvas or in class):
      1. If P represents population and P approaches a constant value M as the time t increases to infinity, then M is called the _____ _____. (There are two answers to this in the text; I'll accept either of them.)
      2. An equilibrium solution M of a differential equation (or system of equations) is _____ if, for any non-equilibrium solution X with an initial value sufficiently close to M, X tends to M as the independent variable t increases to ∞; M is _____ if, for any non-equilibrium solution X with an initial value sufficiently close to M, X → M as t → −∞.
    • Exercises from the textbook due on March 27 Thursday (submit these through MyLab in the Next item): 2.1.1, 2.1.5, 2.1.9, 2.1.17, 2.1.21, 2.1.23, 2.2.1, 2.2.5, 2.2.7.
15. Autonomous systems:
    • Reading from the textbook: Section 6.1; Figure 5.3.16 from Section 5.3 following "A 3-Dimensional Example" (ignore the discussion of matrices and vectors, but see how the eigenvalues determine the shape of the direction field for a linear autonomous system, which are the same shapes as those near the equilibrium solutions of a nonlinear autonomous system).
    • Exercises due on March 27 Thursday (submit these here on Canvas or in class): Suppose that dx/dt = F(x, y) and dy/dt = G(x, y). Fill in the blanks with vocabulary words or mathematical expressions:
      1. Since the independent variable t never appears in it directly, this is a(n) _____ system of differential equations.
      2. Assume that F(a, b) and G(a, b) are not both 0. In forming a direction field for this system of equations, the slope of the arrow through the point (a, b) should be ___ (or infinite if this does not exist).
      3. Now suppose that F(a, b) = 0 and G(a, b) = 0. The solution (x, y) = (a, b) is called a(n) _____ solution of the system.
    • Exercises from the textbook due on April 1 Tuesday (submit these through MyLab in the Next item): 6.1.2, 6.1.5, 6.1.7, 6.1.9, 6.1.23, 6.1.25.
16. Linearization of nonlinear systems:
    • Reading from the textbook: Section 6.2. (Note that the eigenvalues that they are referring to here, when discussing the methods of Chapter 5 that we didn't cover, are the same as the eigenvalues of the second-order equations that you get from solving these systems using the method of Section 4.2.)
    • Exercises due on April 1 Tuesday (submit these here on Canvas or in class): Suppose that you have an autonomous system of two first-order differential equations with an isolated critical point. Consider the eigenvalues of the linearization near that critical point. Identify whether the equilibrium solution associated with that critical point is stable or unstable in the following situations:
      1. The eigenvalues are both positive.
      2. The eigenvalues are both negative.
      3. One eigenvalue is positive, and one is negative.
      4. The eigenvalues are imaginary with a positive real part.
      5. The eigenvalues are imaginary with a negative real part.
    • Exercises from the textbook due on April 3 Thursday (submit these through MyLab in the Next item): 6.2.2, 6.2.3, 6.2.7, 6.2.8, 6.2.10, 6.2.13, 6.2.15, 6.2.16, 6.2.17, 6.2.29, 6.2.31.
17. Interacting population models:
    • Reading from the textbook: Section 6.3.
    • Exercises due on April 3 Thursday (submit these here on Canvas or in class): Suppose that you have a system of equations: dx/dt = F(x) + axy and dy/dt = G(y) + bxy, where t represents time, x and y represent populations of two interacting species, F and G are some functions (describing how the populations would change if the two species never interacted), and a and b are some constants (describing the effects when the two species interact).
      1. If a is positive and b is negative, then which species (x or y) is the predator, and which species (x or y) is the prey?
      2. If the two species are competitors of each other, then will a and b be positive or negative?
      3. If the two species cooperate with each other, then will a and b be positive or negative?
    • Exercises from the textbook due on April 8 Tuesday (submit these through MyLab in the Next item): 6.3.1, 6.3.27, 6.3.29, 6.3.31, 6.3.33.

Laplace transforms

18. Basic Laplace transforms:
    • Reading from the textbook: Section 7.1.
    • Exercises due on April 8 Tuesday (submit these here on Canvas or in class):
      1. Let f be a function defined on (at least) [0, ∞), and let F be the Laplace transform of f (if it exists). Write a formula for F(s) as an integral involving f and s.
      2. Let a be a real number, let f(t) be e^at for t ≥ 0, and let F be the Laplace transform of f. Write a formula for F(s) involving s and a (without an explicit integral).
      3. In the previous question, what is the domain of F?
    • Exercises from the textbook due on April 10 Thursday (submit these through MyLab in the Next item): 7.1.3, 7.1.7, 7.1.13, 7.1.16, 7.1.19, 7.1.23, 7.1.27, 7.1.29.
19. Solving equations with Laplace transforms:
    • Reading from the textbook: Section 7.2.
    • Exercises due on April 10 Thursday (submit these here on Canvas or in class):
      1. If f is a differentiable function of exponential order on [0, ∞) and the Laplace transform of f is F, then what is the Laplace transform of the derivative f⁠′?
      2. If f is a twice differentiable function on [0, ∞), its derivative f⁠′ has exponential order, and the Laplace transform of f is F, then what is the Laplace transform of the second derivative f⁠″?
      3. Put the following steps (which are listed alphabetically) in the order in which you would apply them to solve an initial-value problem:
         1. Solve the equation algebraically;
         2. Take the inverse Laplace transform of both sides of the equation;
         3. Take the Laplace transform of both sides of the equation.
    • Exercises from the textbook due on April 17 Thursday (submit these through MyLab in the Next item): 7.2.1, 7.2.3, 7.2.5, 7.2.8.
20. Inverse transforms of translations:
    • Reading from the textbook: Section 7.3.
    • Exercises due on April 17 Thursday (submit these here on Canvas or in class): Let f be a function defined on [0, ∞), let F be the Laplace transform of f, and let a be a real number.
      1. If g(t) = e^atf(t) whenever t ≥ 0, then what is the Laplace transform of g?
      2. If G(s) = F(s + a) whenever F is defined at s + a, then what is the inverse Laplace transform of G?
      Warning: Pay attention to the signs here!
    • Exercises from the textbook due on April 22 Tuesday (submit these through MyLab in the Next item): 7.3.1, 7.3.3, 7.3.7, 7.3.9, 7.3.13, 7.3.15, 7.3.19, 7.3.27, 7.3.31, 7.3.37.
21. Convolution:
    • Reading from the textbook: Section 7.4.
    • Exercises due on April 22 Tuesday (submit these here on Canvas or in class): Let f and g be continuous functions defined on [0, ∞).
      1. Write a formula for the convolution f ∗ g of f and g.
      2. If f and g have exponential order, F is the Laplace transform of f, and G is the Laplace transform of g, then what is the Laplace transform of f ∗ g?
      3. If f has exponential order, F is the Laplace transform of f, and h(t) = t f(t) whenever t ≥ 0, then what is the Laplace transform of h?
    • Exercises from the textbook due on April 24 Thursday (submit these through MyLab in the Next item): 7.4.3, 7.4.5, 7.4.7, 7.4.15, 7.4.17, 7.4.19, 7.4.21, 7.4.24, 7.4.26, 7.4.29.
22. Discontinuous and periodic forcing:
    • Reading from the textbook: Section 7.5.
    • Exercises due on April 24 Thursday (submit these here on Canvas or in class): Let f be a continuous function defined on [0, ∞), let F be the Laplace transform of f, and let a be a positive real number.
      1. If g(t) = u_a(t) f(t − a) whenever t ≥ 0, where u_a is the unit step function at a, then what is the Laplace transform of g?
      2. If f is periodic with period a, write F(s) for s > 0 as an expression involving a bounded integral.
    • Exercises from the textbook due on April 29 Tuesday (submit these through MyLab in the Next item): 7.5.1, 7.5.3, 7.5.5, 7.5.11, 7.5.13, 7.5.21, 7.5.28, 7.5.33.
23. Impulses:
    • Reading from the textbook: Section 7.6.
    • Exercises due on April 29 Tuesday (submit these here on Canvas or in class): Let a be a positive number.
      1. What is the Laplace transform of the delta function δ_a?
      2. Suppose that a system is undergoing free harmonic motion as described by the differential equation D_t²x + 2pD_tx + ω₀²x = 0 in appropriate units. If the system receives an impulse of amount I at time a (so that it is no longer free), then what differential equation involving a delta function does the entire system satisfy instead?
    • Exercises from the textbook due on May 1 Thursday (submit these through MyLab in the Next item): 7.6.3, 7.6.5, 7.6.7, 7.6.9, 7.6.11.

Quizzes

1. Basics:
   • Review date: February 13&18.
   • Date taken: February 18 Tuesday.
   • Corresponding problems sets: 1–6.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
2. Linear equations:
   • Review date: March 18&20.
   • Date taken: March 20 Thursday.
   • Corresponding problems sets: 7–11.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
3. Systems of equations and applications:
   • Review date: April 10&15.
   • Date taken: April 15 Tuesday.
   • Corresponding problems sets: 12–17.
   • Help allowed: Your notes, calculator.
   • NOT allowed: Textbook, my notes, other people, websites, etc.
4. Laplace transforms:
   • Review date: May 1&6.
   • Date taken: May 6 Tuesday.
   • Corresponding problems sets: 18–23.
   • Help allowed: Your notes, calculator, a table of Laplace transforms provided by me.
   • NOT allowed: Textbook, anything else from my notes, other people, websites, etc.

Final exam

For the exam, you may use one sheet of notes that you wrote yourself. However, you may not use your book or anything else not written by you. You certainly should not talk to other people! Calculators are allowed (although you shouldn't really need one), but not communication devices (like cell phones).

The exam consists of questions similar in style and content to those in the practice exam (DjVu).

The permanent URI of this web page is https://tobybartels.name/MATH-2200/2025SP/.