Chapter 8 - Differential Equations

📚 Table of Contents

1. [Slope Fields]
2. [Separable Equations]
3. [Logistic Growth]
4. [Euler's Method]
5. [Applications]

Basic Concepts 📈

Key Terms

• Order: Highest derivative
• Linear vs. Nonlinear
• Initial value problem (IVP)
• General solution

Common Forms

1. Separable: $\frac{dy}{dx} = f(x)g(y)$
2. Linear: $\frac{dy}{dx} + P(x)y = Q(x)$
3. Logistic: $\frac{dy}{dt} = ky(1-\frac{y}{M})$

Solution Methods 🔧

Separation of Variables

1. Separate y and x terms
2. Integrate both sides
3. Solve for y
4. Apply initial conditions

Integration

$y = \int f(x)dx + C$

Euler's Method

$y_{n+1} = y_n + f(x_n,y_n)\Delta x$

Applications 🎯

Growth/Decay

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

• Solution: $y = Ce^{kt}$

Logistic Growth

$\frac{dy}{dt} = ky(1-\frac{y}{M})$

• Solution: $y = \frac{M}{1 + Ce^{-kt}}$

Newton's Law of Cooling

$\frac{dT}{dt} = k(T-T_s)$

• Solution: $T = T_s + (T_0-T_s)e^{kt}$

1. Slope Fields 📊

Understanding Slope Fields

A visual representation of differential equations. Think of it as:

• Direction field for solutions
• Map of solution curves
• Tangent line indicators
• Solution path guide

Creating Slope Fields

Process

1. Identify dy/dx:

   • Write equation in standard form
   • Isolate derivative
   • Understand variables
   • Note special cases

2. Plot Slopes:

   • Choose grid points
   • Calculate slope at each point
   • Draw short line segments
   • Maintain consistent length

3. Analyze Pattern:

   • Look for symmetry
   • Find equilibrium solutions
   • Note key behaviors
   • Identify solution types

Example Walkthrough - Slope Fields

For $\frac{dy}{dx} = x - y$

1. Create grid:

   • Choose points (-2,-2) to (2,2)
   • Mark intersections
   • Consider scale
   • Note key points

2. Calculate slopes:

   • At (0,0): slope = 0
   • At (1,0): slope = 1
   • At (0,1): slope = -1
   • Pattern emerges

3. Draw segments:

   • Use consistent length
   • Show direction
   • Connect smoothly
   • Verify pattern

2. Separable Equations 📈

Understanding Separable Equations

Equations where variables can be separated. Think of it as:

• Grouping like terms
• Dividing variables
• Independent integration
• Reverse chain rule

Solution Process

Steps for Solving

1. Identify Form:

   • Check if $\frac{dy}{dx} = f(x)g(y)$
   • Variables must separate
   • No mixed terms
   • Clear denominators

2. Separate Variables:

   • Move y terms to one side
   • Move x terms to other side
   • Write in differential form
   • Check domain

3. Integrate Both Sides:

   • Use standard integrals
   • Watch for substitutions
   • Include constants
   • Check signs

4. Solve for y:

   • Isolate y variable
   • Use algebra
   • Consider domain
   • Verify solution

Example Walkthrough

Solve $\frac{dy}{dx} = xy$

1. Separate:
   • $\frac{dy}{y} = xdx$
2. Integrate:
   • $\ln|y| = \frac{x^2}{2} + C$
3. Solve:
   • $y = ±e^{x^2/2 + C}$
   • $y = Ce^{x^2/2}$

3. Logistic Growth 📊

Understanding Logistic Growth

Models bounded growth. Think of it as:

• Limited population growth
• S-shaped curve
• Carrying capacity model
• Restricted exponential growth

Standard Form

$\frac{dy}{dt} = ky(1-\frac{y}{M})$

Components

1. Growth Rate (k):

   • Initial growth speed
   • Positive constant
   • Affects steepness
   • Time scaling

2. Carrying Capacity (M):

   • Maximum value
   • Horizontal asymptote
   • Limiting factor
   • Final equilibrium

Solution Process

1. Set Up Equation:

   • Identify k and M
   • Write in standard form
   • Check initial conditions
   • Note constraints

2. Separate Variables:

   • Partial fractions
   • Split terms
   • Integrate both sides
   • Solve for y

3. Final Form:

   • $y = \frac{M}{1 + Ce^{-kt}}$
   • C from initial condition
   • Verify solution
   • Check behavior

Example Walkthrough

Population with k = 0.5, M = 1000, P₀ = 100

1. Write equation:
   • $\frac{dP}{dt} = 0.5P(1-\frac{P}{1000})$
2. Use formula:
   • $P = \frac{1000}{1 + Ce^{-0.5t}}$
3. Find C:
   • Use P(0) = 100
   • Solve for C = 9

4. Euler's Method 🔄

Understanding Euler's Method

Numerical approximation technique. Think of it as:

• Step-by-step approximation
• Tangent line walking
• Numerical integration
• Local linear approximation

Formula

$y_{n+1} = y_n + f(x_n,y_n)\Delta x$

Implementation Process

1. Set Up:

   • Choose Δx
   • List initial values
   • Create table
   • Plan steps

2. Calculate Steps:

   • Find slope
   • Use formula
   • Record values
   • Continue pattern

3. Analyze Error:

   • Smaller Δx = better
   • Accumulating error
   • Compare to actual
   • Consider improvements

Example Walkthrough

Use Euler's method for $\frac{dy}{dx} = x + y$, y(0) = 1, Δx = 0.2

1. Create table:
   • x₀ = 0, y₀ = 1
2. First step:
   • f(0,1) = 0 + 1 = 1
   • y₁ = 1 + 1(0.2) = 1.2
3. Continue:
   • x₁ = 0.2, y₁ = 1.2
   • f(0.2,1.2) = 1.4
   • y₂ = 1.2 + 1.4(0.2)

5. Applications 🎯

Types of Applications

1. Population Growth:

   • Exponential
   • Logistic
   • Predator-prey
   • Competition

2. Decay Problems:

   • Radioactive decay
   • Temperature
   • Half-life
   • Cooling

3. Financial Models:

   • Compound interest
   • Investment growth
   • Payment schedules
   • Market models

Example Walkthrough

Newton's Law of Cooling: T(0) = 100°, room 70°

1. Write equation:
   • $\frac{dT}{dt} = k(T-70)$
2. Solve:
   • $T = 70 + Ce^{kt}$
3. Use initial condition:
   • 100 = 70 + C
   • C = 30

📝 AP-Style Examples

Example 1: Slope Fields

Given $\frac{dy}{dx} = x^2 - y$, sketch slope field and solution curve through (0,1)

Solution:

1. Create slope field:

   • Calculate slopes at key points
   • Draw segments carefully
   • Note equilibrium curve y = x²
   • Show solution direction

2. Solution curve:

   • Start at (0,1)
   • Follow slope pattern
   • Note curve approaches y = x²
   • Show decreasing initially

Example 2: Euler's Method

Use two steps of Euler's method to approximate y(0.4) for $\frac{dy}{dx} = x + y$, y(0) = 1, Δx = 0.2

Solution:

1. First step:

   • y₁ = 1 + (0 + 1)(0.2)
   • y₁ = 1.2 at x = 0.2

2. Second step:

   • y₂ = 1.2 + (0.2 + 1.2)(0.2)
   • y₂ = 1.48 at x = 0.4

Example 3: Logistic Growth

A population follows $\frac{dP}{dt} = 0.1P(1-\frac{P}{500})$ with P(0) = 50. Find P(10).

Solution:

1. Identify:

   • k = 0.1
   • M = 500
   • P₀ = 50

2. Use formula:

   • $P = \frac{500}{1 + 9e^{-0.1t}}$
   • Substitute t = 10
   • P(10) ≈ 187.8

💡 Success Strategies

1. Equation Classification

• Check for separable form first
• Look for linear patterns
• Identify growth/decay type
• Consider special cases

2. Solution Process

• Draw slope field first
• Use Euler's for approximation
• Solve analytically when possible
• Verify with initial conditions

3. Common Mistakes

1. Slope Fields:

   • Wrong slope direction
   • Inconsistent segment length
   • Missing key points
   • Incorrect equilibrium lines

2. Euler's Method:

   • Wrong step size
   • Calculation errors
   • Not showing work
   • Skipping intermediate values

3. Analytical Solutions:

   • Incorrect separation
   • Missing absolute value
   • Wrong integration
   • Forgetting +C

🔍 AP Exam Focus

Free Response Tips

1. Slope Fields:

   • Show clear segments
   • Consistent length
   • Correct direction
   • Solution curves match field

2. Euler's Method:

   • Organize in table
   • Show all calculations
   • Round appropriately
   • Include units

3. Analytical Solutions:

   • Show separation clearly
   • Include all steps
   • Verify initial conditions
   • Check final answer

Multiple Choice Strategy

1. Quick Checks:

   • Direction field pattern
   • Solution behavior
   • Initial condition match
   • Long-term behavior

2. Common Traps:

   • Sign errors
   • Missing factors
   • Wrong method choice
   • Incomplete solutions

📊 Quick Reference

Equation Types

1. Separable: $\frac{dy}{dx} = f(x)g(y)$
2. Linear: $\frac{dy}{dx} + P(x)y = Q(x)$
3. Logistic: $\frac{dy}{dt} = ky(1-\frac{y}{M})$

Key Methods

1. Separation:

   • $\int \frac{dy}{g(y)} = \int f(x)dx$
   • Solve for y
   • Use initial conditions

2. Euler's:

   • $y_{n+1} = y_n + f(x_n,y_n)\Delta x$
   • Table format
   • Multiple steps

3. Slope Fields:

   • Grid points
   • Short segments
   • Consistent length
   • Show pattern

Important Solutions

1. Exponential Growth/Decay:

   • $y = Ce^{kt}$
   • Unrestricted growth
   • Proportional change

2. Logistic Growth:

   • $y = \frac{M}{1 + Ce^{-kt}}$
   • Limited growth
   • S-shaped curve

3. Linear:

   • Use integrating factor
   • $μ(x) = e^{\int P(x)dx}$
   • Multiply through

💡 Pro Tip: When in doubt, start with a slope field - it gives you the big picture of solution behavior!