Unit 7: Differential Equations

Master the art of solving differential equations and modeling real-world phenomena

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📋 Unit Overview

Unit 7 focuses on differential equations and their applications. You'll learn how to:

  • Solve separable differential equations
  • Use slope fields to visualize solutions
  • Apply Euler's method for approximation
  • Model exponential growth and decay
  • Solve logistic growth problems
💡 Key Insight: Differential equations describe how quantities change over time. They're essential for modeling population growth, radioactive decay, and many other real-world phenomena.

🔍 Introduction to Differential Equations

What is a Differential Equation?

A differential equation is an equation that relates a function with its derivatives. The general form is:

F(x, y, y', y'', ..., y⁽ⁿ⁾) = 0

where y' = dy/dx, y'' = d²y/dx², etc.

🎯 Key Concept

Differential equations describe how a quantity changes in relation to itself and other variables.

📝 Example 7.1: Basic Differential Equation

dy/dx = 2x

Solution: y = ∫2x dx = x² + C

This is a family of parabolas, each with a different y-intercept.

Types of Differential Equations

Type Form Example
Separable dy/dx = f(x)g(y) dy/dx = xy
Linear dy/dx + P(x)y = Q(x) dy/dx + 2y = x
Homogeneous dy/dx = f(y/x) dy/dx = (x+y)/(x-y)

🔧 Separable Differential Equations

A separable differential equation can be written in the form:

dy/dx = f(x)g(y)

We can separate variables and integrate both sides

🔑 Solution Method
  1. Separate variables: dy/g(y) = f(x)dx
  2. Integrate both sides: ∫dy/g(y) = ∫f(x)dx
  3. Solve for y: Express y as a function of x
📝 Example 7.2: Separable Equation

Solve: dy/dx = xy

Solution:

  1. Separate: dy/y = x dx
  2. Integrate: ∫dy/y = ∫x dx
  3. ln|y| = x²/2 + C
  4. y = e^(x²/2 + C) = Ce^(x²/2)

📈 Slope Fields

Understanding Slope Fields

A slope field (or direction field) is a graphical representation of a differential equation. At each point (x,y), we draw a small line segment with slope equal to dy/dx at that point.

🎯 Key Concept

Slope fields help us visualize the family of solutions to a differential equation without solving it analytically.

📝 Example 7.3: Creating a Slope Field

For dy/dx = x - y:

  • At (0,0): slope = 0 - 0 = 0 (horizontal line)
  • At (1,0): slope = 1 - 0 = 1 (45° line)
  • At (0,1): slope = 0 - 1 = -1 (downward 45° line)

Draw small line segments at each point with these slopes.

Using Slope Fields

1. Sketch Solutions

Follow the slope field to sketch approximate solution curves.

2. Find Particular Solutions

Use initial conditions to find specific solution curves.

📝 Example 7.4: Particular Solution

Given dy/dx = x - y and y(0) = 1, sketch the solution curve.

Solution: Start at (0,1) and follow the slope field to sketch the curve that passes through this point.

🔢 Euler's Method

Euler's method is a numerical technique for approximating solutions to differential equations:

yₙ₊₁ = yₙ + h · f(xₙ, yₙ)

where h is the step size

🔑 Algorithm Steps
  1. Start with initial condition: (x₀, y₀)
  2. Choose step size: h
  3. Calculate next point: x₁ = x₀ + h, y₁ = y₀ + h·f(x₀, y₀)
  4. Repeat: Continue the process
📝 Example 7.5: Euler's Method

Approximate y(1) for dy/dx = x + y, y(0) = 1, using h = 0.5

Solution:

  • Step 1: x₁ = 0.5, y₁ = 1 + 0.5(0 + 1) = 1.5
  • Step 2: x₂ = 1.0, y₂ = 1.5 + 0.5(0.5 + 1.5) = 2.5

So y(1) ≈ 2.5

📊 Exponential Growth and Decay

Exponential Growth

The differential equation for exponential growth is:

dy/dt = ky

where k > 0 is the growth constant

📝 Example 7.6: Population Growth

A population grows at a rate proportional to its size. If P(0) = 1000 and P(2) = 2000, find P(t).

Solution:

  1. dP/dt = kP
  2. Separate: dP/P = k dt
  3. Integrate: ln|P| = kt + C
  4. P = Ce^(kt)
  5. Use P(0) = 1000: C = 1000
  6. Use P(2) = 2000: 2000 = 1000e^(2k), so k = ln(2)/2
  7. P(t) = 1000e^(t·ln(2)/2) = 1000·2^(t/2)

Exponential Decay

For exponential decay, the equation is the same but k < 0:

y = y₀e^(kt)

where k < 0 and y₀ is the initial amount

📝 Example 7.7: Radioactive Decay

A radioactive substance decays at a rate proportional to its mass. If 50% remains after 10 years, find the half-life.

Solution:

  1. y = y₀e^(kt)
  2. 0.5y₀ = y₀e^(10k), so k = ln(0.5)/10
  3. For half-life: 0.5y₀ = y₀e^(kt), so t = ln(0.5)/k = 10 years

📈 Logistic Growth

Logistic growth models situations where growth is limited by carrying capacity:

dy/dt = ky(L - y)

where L is the carrying capacity and k > 0

🎯 Key Features
  • Carrying capacity: Maximum sustainable population
  • S-shaped curve: Slow growth, then rapid, then slow again
  • Realistic model: Accounts for limited resources
📝 Example 7.8: Logistic Growth

Solve dy/dt = 0.1y(100 - y) with y(0) = 10

Solution:

  1. Separate: dy/[y(100-y)] = 0.1 dt
  2. Use partial fractions: 1/[y(100-y)] = A/y + B/(100-y)
  3. Integrate and solve: y = 100/(1 + 9e^(-10t))

🧮 Practice Problems

Problem 1

Solve the separable differential equation: dy/dx = x²y

Solution:

1) Separate variables: dy/y = x² dx

2) Integrate both sides: ∫dy/y = ∫x² dx

3) ln|y| = x³/3 + C

4) y = e^(x³/3 + C) = Ce^(x³/3)

Problem 2

Use Euler's method with h = 0.5 to approximate y(1) for dy/dx = 2x + y, y(0) = 1.

Solution:

Step 1: x₁ = 0.5, y₁ = 1 + 0.5(2(0) + 1) = 1 + 0.5 = 1.5

Step 2: x₂ = 1.0, y₂ = 1.5 + 0.5(2(0.5) + 1.5) = 1.5 + 0.5(2.5) = 2.75

Therefore, y(1) ≈ 2.75

Problem 3

A bacteria culture grows exponentially. If it doubles every 3 hours and starts with 100 bacteria, find the population after 6 hours.

Solution:

1) P(t) = P₀e^(kt) where P₀ = 100

2) P(3) = 200 = 100e^(3k), so e^(3k) = 2

3) k = ln(2)/3

4) P(6) = 100e^(6·ln(2)/3) = 100e^(2ln(2)) = 100·2² = 400

After 6 hours, there are 400 bacteria.

💡 Exam Tips

🎯 Multiple Choice
  • Look for separable equations first
  • Check if you can use slope fields
  • Remember exponential growth/decay formulas
  • Use Euler's method for approximations
✍️ Free Response
  • Show all steps in separation of variables
  • Include initial conditions in your solution
  • Sketch slope fields carefully
  • Interpret your results in context
⚠️ Common Mistakes:
  • Forgetting the constant of integration
  • Not applying initial conditions correctly
  • Confusing growth and decay constants
  • Making errors in Euler's method calculations

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➡️ What's Next?

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In Unit 8, you'll learn about applications of integration including area, volume, and arc length.

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