MA266 Fall 2026 : Lesson 1 (08/24/26)

MA266 Fall 2026 : Lesson 1 (08/24/26)

Today:

• Syllabus discussion
• Course introduction
• Introduction to Differential equations and Mathematical Models (1.1)

Announcements:

• Read Syllabus
• Set up my lab math

Office Hours: MWF: 1:30 PM – 2:30 PM, MATH 842

What is a differential equation?

differential: change

Equation: Statement that values of two mathematical expressions are equal

eg:

• \(\sin x = 1\)
• \(2 + 2 = 4\)
• \(x^2 + 5y = z\)

DE: equation involving an unknown function and its derivatives

eg:

\[ y' = e^x \] \[ y'' + 4y' + y = \sin x \] \[ y' = 1 \]

• \(y\) is fn of 1 variable
• ODE (ordinary DE)

\[ \frac{\partial z}{\partial y} + \frac{\partial z}{\partial x} + xyz = 0 \]

• \(z\) is a function of 2 variables
• PDE

Which of the following are ODEs

1. \( f'(x) = f(x) + \sin x \) ✓

2. \( f(x) = \sin x \) ✗

3. \( \sin x = 1 \) ✗

4. \( f'(x) + 3f(x) \le 0 \) ✗

5. \( g''(x) + 2g'(x) + g(x) = 5\cos(x) \) ✓

6. \( \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} = x^2 + y^2 \) ✗ (PDE)

Why do we want to study differential equations?

The modeling cycle consists of the following stages:

• Real-World Systems / Phenomena:

  • weather
  • infectious disease
  • stock market
  • water flow
  • physical

• Step 1: Model → Formulate a differential equation from the physical system.

• Step 2: Solve → Find the solution to the differential equation.

  (Note: Most of MA266 focuses on this step.)

• Step 3: Interpret → Interpret the solution back in the context of the original real-world problem.

eg: Logistic model for an infectious disease

• \(I(t)\): No of people infected
• \(\gamma\): rate of spread
• \(P\): total population

Model \(\longrightarrow\)

\[ I'(t) = r I(P - I) \]

Solve:
Later in MA266

\[ I(t) = \frac{P I_0}{I_0 + (P - I_0) e^{-r P t}} \]

Interpret:

Interpretation: In long run (\(t \to \infty\)) \(\to \lim_{t \to \infty} I(t) = P\)

\(\Rightarrow\) everyone will be infected

Order of a differential equation

→ highest derivative that is involved in the ODE

eg:

• \( y' = e^x \rightsquigarrow 1^{\text{st}} \text{ order ODE} \)
• \( y'' + 9y' - 7y = \sin x \rightsquigarrow 2^{\text{nd}} \text{ order ODE} \)
• \( y''' + (y'')^2 + y' \cdot y = 0 \rightsquigarrow 3^{\text{rd}} \text{ order ODE} \)
• \( (y^{(5)})^2 + 3 y^{(4)} \cdot y'' - 8 \sin x = 0 \rightsquigarrow 5^{\text{th}} \text{ order ODE} \)

\( n\text{th} \text{ order ODE} \rightsquigarrow F(x, y, y', y'', \dots, y^{(n)}) = 0 \)

Solution of a differential equation

A function that satisfies the ODE in an interval.

Just like \( x = 1 \), \( x = 2 \) are solutions to

\[ x^2 - 3x + 2 = 0 \]

eg: \( y = e^x + 7 \) is a solution to \( y' = e^x \)

\[ \frac{d}{dx} \left[ e^x + 7 \right] = e^x \]

* \( y = e^x + C \) is a "general" solution.

eg: \( y = \frac{x^3}{3} + \frac{1}{x} \) is a solution to \( xy' + y = \frac{4x^3}{3} \) on \( \mathbb{R} \setminus \{0\} \)

\[ \frac{d}{dx} \left[ \frac{x^3}{3} + \frac{1}{x} \right] = x^2 - x^{-2}, \quad xy' + y = x(x^2 - x^{-2}) + \frac{x^3}{3} + \frac{1}{x} = \frac{4x^3}{3}. \]

Initial value Problem (IVP)

finding solution to ODE with an initial condition

eg: \( y' = e^x, \quad y(0) = 8 \)

\( \downarrow \) gen. solution

\[ y = e^x + k \xrightarrow{\text{plug in initial cond.}} y(0) = 8 = 1 + k \implies k = 7 \]

\[ y(x) = e^x + 7 \]

eg: \( y' = x^3, \quad y(0) = -3 \)

\( \downarrow \) gen solution

\[ y = \frac{x^4}{4} + k \xrightarrow[\text{plug in } y(0) = -3]{} -3 = 0 + k \implies k = -3 \quad \left\} \quad y(x) = \frac{x^4}{4} - 3 \right. \]