Textbook (Jiří Lebl's Notes on Diffy Q's)
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Mathematica Tutorial (Mathematica Notebook)
St. Andrews History of Mathematics Archive
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Chapter 1: Definitions, Families of Curves
1.1: Examples of Differential Equations
1.2: Definitions
Read 0.2.1, 0.2.2, and 0.3 in Jiří Lebl's Notes on Diffy Q's (Lebl)
Partial Derivatives (Functions of Two or Three Variables, Partial Derivatives (Brief), Partial Derivatives (Detailed))
HW # 1: Lebl 0.2.5 Exercises #0.2.101 - 0.2.104; 0.3.1 Exercises #0.3.101, 0.3.103 Also, for each problem, identify any independent variables, dependent variables, and parameters.
1.3: Families of Solutions
1.4: Geometric Interpretation
Example 4, 5, and 6 (Mathematica Notebook)
Read 0.2.3 and 0.2.4 in Lebl
Read 1.1 in Lebl
HW # 2: 1.3 Handout Problems: # 1, 3, 5, 7, 9, 12, 13, 14. 1.4 Handout Problems: Do what it says in #1, #2 for exercises 1, 3, 5, 7, 9 in 1.3 Handout Problems. Eliminate the constants to find the differential equation associated with y = A(x-B)2 (Hint: Differentiate twice.). If a relation has n constants (or parameters) that are eliminated, what would you expect about the resulting differential equation? Lebl 0.2.5 Exercises # 0.2.13, 0.2.106, Lebl 1.1.1 Exercises # 1.1.106, 1.1.107.
1.5: Method of Isoclines; Direction Fields
Direction Fields via Mathematica (Mathematica Notebook)
Direction Fields via John C. Polking’s dfield program (web-based java version of dfield)
Direction Fields via (simplified web-page version of dfield)
1.6: An Existence and Uniqueness Theorem
Read 1.2.2 in Lebl.
HW # 3: Lebl 1.2.3 Exercises (Use technology for drawing direction fields.) 1.2.2, 1.2.3, 1.2.4, 1.2.6, 1.2.101, 1.2.102, 1.2.103, 1.2.104, 1.2.106. Apply the Method of Isoclines to # 1.2.2, 1.2.3, 1.2.6.
Chapter 2: Equations of Order One
2.1: Separation of Variables
Example 3 Plot (Mathematica)
Partial Fraction Decomposition Examples (Mathematica Notebook)
Partial Fractions Handout (PDF)
Read 1.3, 1.3.1, 1.3.2, and 1.3.3 in Lebl.
HW # 4: Lebl 1.3.4 Exercises # 1.3.2, 1.3.3, 1.3.4, 1.3.5, 1.3.6, 1.3.7, 1.3.8, 1.3.11, 1.3.102, 1.3.106, 1.3.107
2.2: Homogeneous Functions
2.3: Equations with Homogeneous Coefficients
Implicit Plots with Mathematica (Mathematica Notebook)
Read 1.5.3 in Lebl
HW # 5: Lebl 1.5.4 Exercises # 1.5.4, 1.5.6, 1.5.102. Handout Problems # 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 22, 36.
2.4: Exact Equations
Review of Differentials Handout (PDF); More on Differentials (see pages 5-10) and the Total Differential (see pages 10-12)
Exact Equations Handout (PDF)
Rest of Theorem 2.3 Proof (PDF)
Read 1.8, 1.8.1, and 1.8.2 in Lebl
HW # 6: Lebl 1.8.3 Exercises # 1.8.101, 1.8.102(d), 1.8.103; For 1.8.102(d), solve the resulting exact equation. Can you think of another way to solve 1.8.102(d), based on what we have seen in class? Fill in any missing details in the proof of Theorem 2.3 (Hint: Use Clairaut’s Theorem, the Fundamental Theorem of Calculus, and Fubini’s Theorem.)
2.5: Linear Equations of Order One
Read 1.4 in Lebl
HW # 7: Lebl 1.4.1 Exercises # 1.4.4, 1.4.8, 1.4.10, 1.4.102, 1.4.103, 1.4.105
2.6: How to Solve Differential Equations with Mathematica
Mathematica Examples (Mathematica Notebook)
Chapter 3: Elementary Applications
3.1: Velocity of Escape from the Earth
3.2: Simple Chemical Conversion (Radioactive Decay)
3.3: Logistic Growth
Read Section 2.4 Applications (2.4.1 - 2.4.6) in Jeffrey R. Chasnov's Differential Equations
HW #8: Lebl 0.2.5 Exercises # 0.2.14; 1.1.1 Exercises # 1.1.11; 1.3.4 Exercises # 1.3.12; 1.4.1 Exercises # 1.4.104. Use equation (6) from section 3.1 of Lecture 8, along with data from this HyperPhysics Escape Velocity page to show that the escape velocity of Mercury is approximately 4.3 km/sec. Find the general solution of the logistic equation (3) from section 3.3 of Lecture 8, then using the initial condition x(0) = x0, show that particular solution (5) results.
3.4: Autonomous Equations and Phase Lines
Read 1.6 in Lebl
HW # 9: Lebl 1.6.1 Exercises # 1.6.101, 1.6.102, 1.6.103, 1.6.104. Use the ideas of example 2 in our section 3.4 class notes to investigate the behavior of the differential equations y’ = k y for k>0 or k<0 and y’ = k(y-T) for a fixed constant T with k>0 or k<0.
HW # 10: Handout problems # 1-5 on the Logistic Equation (use a = 2 and b = 3 in all problems) (web-based java version of dfield) (simplified web-page version of dfield)
Midterm Exam 1 will be posted on Wednesday, 9/24/2025 at 8:00 am and due on Thursday, 9/25/2025 at 11:59 pm. It will cover all material up to and including Lecture 9 and HW # 10. The exam will consist of two parts, (1) a timed portion for which (once opened) you will be allocated 180 minutes to complete and submit by the 11:59 pm deadline; (2) a “take-home” portion to be submitted by the 11:59 pm deadline. Due to exam time constraints, I suggest you prepare personal notes to be able to efficiently complete the exam in the time allowed.
(Optional) Homework #1 - #10 is due on Wednesday, 9/24/2025 at 11:59 pm. Please submit your work via the Canvas link provided for this assignment.
Chapter 4: Linear Differential Equations
4.1: The General Linear Equation
4.2: An Existence and Uniqueness Theorem
HW # 11: Lebl 2.1.1 Exercises # 2.1.4, 2.1.5, 2.1.6, 2.1.7, 2.1.104, 2.1.105
4.3: Linear Independence
Read Notes on Linear Independence from Boyce and DiPrima's Elementary Differential Equations and Boundary Value Problems (B&D 8th edition) (pp. 153-154): (PDF)
Read 2.3.1 in Lebl
HW # 12: 4.1 & 4.2 Handout Problems # 1-6, 9; B&D 8th edition p. 158 # 1, 2, 5 (PDF); Lebl 2.1.1 Exercises # 2.1.101, 2.1.102; Lebl 2.3.3 Exercises # 2.3.104
4.4: The Wronskian