Differential Equations Intro

🌊 Differential Equations: The Language of Change

Imagine you’re watching a river flow. The water moves, speeds up, slows down, swirls around rocks. How would you describe this dance of motion? That’s exactly what differential equations do—they speak the language of change.

🎯 What You’ll Learn

• What differential equations are (and why they’re everywhere!)
• How to solve separable equations (the friendly ones)
• First-order linear equations (with a magic trick)
• Exponential growth and decay (rabbits and radioactive rocks)

🌟 Chapter 1: What Is a Differential Equation?

The Big Idea

Think of a differential equation like a recipe for change.

When you bake cookies, a recipe tells you: “Mix flour, add sugar, bake at 350°F.”

A differential equation tells you: “Here’s how something changes over time.”

A Simple Story

Meet Lily the Ladybug. 🐞

Lily is climbing up a flower stem. The higher she goes, the faster she moves (she gets excited!).

We can write: “Lily’s speed depends on her height.”

In math language: $\frac{dy}{dt} = ky$

Where:

• y = Lily’s height
• t = time
• dy/dt = how fast height changes (her speed!)
• k = how excited she gets

That’s a differential equation! It connects how something changes (dy/dt) to where it currently is (y).

Real Examples Around You

The Key Vocabulary

graph TD
    A["Differential Equation"] --> B["Contains derivatives"]
    A --> C["Describes change"]
    B --> D["dy/dt, y', etc."]
    C --> E["Rate depends on current state"]

Derivative = How fast something changes Differential equation = An equation with a derivative in it

🧩 Chapter 2: Separable Equations — The Friendly Ones

What Makes Them Separable?

Some equations are like neat roommates—they let you put all the y-stuff on one side and all the t-stuff on the other.

Example: $\frac{dy}{dt} = y \cdot t$

Can be separated into: $\frac{dy}{y} = t \cdot dt$

Now y is on the left, t is on the right. Perfect!

The Recipe 🍳

Step 1: Get dy and y on one side Step 2: Get dt and t on the other side Step 3: Put ∫ (integral) in front of both sides Step 4: Solve!

Worked Example

Problem: Solve $\frac{dy}{dt} = 3y$

Story: This is like saying “The more money you have, the faster it grows!”

Solution:

Step 1: Separate
dy/y = 3dt

Step 2: Integrate both sides
∫(1/y)dy = ∫3dt

Step 3: Solve
ln|y| = 3t + C

Step 4: Get y alone
y = e^(3t + C)
y = e^C · e^(3t)
y = Ae^(3t)  [where A = e^C]

Answer: $y = Ae^{3t}$

Visual Understanding

graph TD
    A["dy/dt = f#40;y#41;·g#40;t#41;"] --> B["Separate: dy/f#40;y#41; = g#40;t#41;dt"]
    B --> C["Integrate: ∫dy/f#40;y#41; = ∫g#40;t#41;dt"]
    C --> D["Solve for y"]
    D --> E["Add constant C"]

Another Example

Problem: Solve $\frac{dy}{dx} = xy$ with $y(0) = 2$

Separate: $\frac{dy}{y} = x , dx$

Integrate: $\ln|y| = \frac{x^2}{2} + C$

Solve: $y = Ae^{x^2/2}$

Use initial condition $y(0) = 2$: $2 = Ae^0 = A$

Final Answer: $y = 2e^{x^2/2}$

🪄 Chapter 3: First-Order Linear Equations

A New Challenge

Not all equations separate nicely. Some look like this: $\frac{dy}{dt} + P(t)y = Q(t)$

This is called a first-order linear equation.

The Magic Trick: Integrating Factor

Imagine you have a locked door. The integrating factor is the special key.

The key (integrating factor) is: $\mu(t) = e^{\int P(t)dt}$

Why Does This Work? (The Simple Version)

When you multiply both sides by this magic $\mu(t)$, something amazing happens—the left side becomes a perfect derivative!

$\mu \cdot \frac{dy}{dt} + \mu \cdot P(t) \cdot y = \frac{d}{dt}[\mu \cdot y]$

Now you can just integrate!

Step-by-Step Recipe

Step 1: Write in standard form: $\frac{dy}{dt} + P(t)y = Q(t)$ Step 2: Find $\mu = e^{\int P(t)dt}$ Step 3: Multiply everything by $\mu$ Step 4: Left side becomes $\frac{d}{dt}[\mu y]$ Step 5: Integrate both sides Step 6: Solve for y

Worked Example

Problem: Solve $\frac{dy}{dt} + 2y = 6$

Step 1: Already in standard form! P(t) = 2, Q(t) = 6

Step 2: Find integrating factor $\mu = e^{\int 2dt} = e^{2t}$

Step 3: Multiply both sides by $e^{2t}$ $e^{2t}\frac{dy}{dt} + 2e^{2t}y = 6e^{2t}$

Step 4: Left side is $\frac{d}{dt}[e^{2t}y]$ $\frac{d}{dt}[e^{2t}y] = 6e^{2t}$

Step 5: Integrate $e^{2t}y = 3e^{2t} + C$

Step 6: Solve for y $y = 3 + Ce^{-2t}$

Visual Flow

graph TD
    A["dy/dt + P#40;t#41;y = Q#40;t#41;"] --> B["Find μ = e^∫P#40;t#41;dt"]
    B --> C["Multiply by μ"]
    C --> D["d/dt[μy] = μQ"]
    D --> E["Integrate both sides"]
    E --> F["Solve for y"]

🐰 Chapter 4: Exponential Growth and Decay

The Universal Pattern

Here’s a secret: so many things in nature follow the same pattern!

$\frac{dy}{dt} = ky$

• If k > 0: Things GROW (rabbits, bacteria, money)
• If k < 0: Things SHRINK (radioactive atoms, medicine in blood)

The Solution (Always!)

$y = y_0 e^{kt}$

Where:

• $y_0$ = starting amount
• $k$ = growth/decay rate
• $t$ = time
• $e$ ≈ 2.718… (nature’s favorite number!)

🐰 Growth Story: The Rabbit Problem

Setup: You start with 100 rabbits. They double every year.

Finding k: If population doubles, $2y_0 = y_0 e^{k \cdot 1}$ $2 = e^k$ $k = \ln(2) \approx 0.693$

The equation: $y = 100e^{0.693t}$

☢️ Decay Story: Radioactive Carbon

Setup: Carbon-14 has a half-life of 5,730 years.

Finding k: Half remains means $0.5y_0 = y_0 e^{k \cdot 5730}$ $0.5 = e^{5730k}$ $\ln(0.5) = 5730k$ $k = \frac{-0.693}{5730} \approx -0.000121$

The equation: $y = y_0 e^{-0.000121t}$

Half-Life Formula

For decay, the half-life ($t_{1/2}$) is: $t_{1/2} = \frac{\ln(2)}{|k|} = \frac{0.693}{|k|}$

Doubling Time Formula

For growth, the doubling time ($t_d$) is: $t_d = \frac{\ln(2)}{k} = \frac{0.693}{k}$

Visual: Growth vs Decay

graph LR
    A["Start"] --> B{k positive?}
    B -->|Yes| C["📈 GROWTH"]
    B -->|No| D["📉 DECAY"]
    C --> E["y → ∞ as t → ∞"]
    D --> F["y → 0 as t → ∞"]

Real-World Applications Table

🎓 The Big Picture

graph TD
    A["Differential Equations"] --> B["Separable"]
    A --> C["First-Order Linear"]
    B --> D["Separate, Integrate, Solve"]
    C --> E["Integrating Factor Method"]
    B --> F["Exponential Growth/Decay"]
    F --> G["y = y₀e^kt"]
    G --> H{k > 0?}
    H -->|Yes| I["Growth"]
    H -->|No| J["Decay"]

🔑 Key Takeaways

1. Differential equations describe change — how things evolve over time

2. Separable equations can be split:

   • Put y’s on one side, t’s on the other
   • Integrate both sides

3. First-order linear equations use a magic trick:

   • Multiply by integrating factor $\mu = e^{\int P(t)dt}$
   • Left side becomes a perfect derivative

4. Exponential growth/decay follows $y = y_0 e^{kt}$:

   • k > 0 means growth
   • k < 0 means decay
   • Half-life = $\frac{0.693}{|k|}$

💪 You’ve Got This!

Differential equations might seem scary at first, but they’re just recipes for change. Nature speaks in differential equations—now you can understand her language!

Remember:

• Start simple (can I separate it?)
• If not, try the integrating factor
• Growth and decay are the same equation—just different signs!

Next time you see something growing or shrinking, you’ll know the math behind it. 🚀