IB Mathematics AI – Topic 5

Definition:

A differential equation is an equation involving a function and its derivatives

General form: relates \(y\), \(\frac{dy}{dx}\), \(\frac{d^2y}{dx^2}\), etc.

Order of Differential Equation:

• First order: Contains only \(\frac{dy}{dx}\) (no higher derivatives)
• Second order: Contains \(\frac{d^2y}{dx^2}\)

Types of Solutions:

• General solution: Contains arbitrary constant(s)
• Particular solution: Uses initial conditions to find specific constant values

Common Notation:

\(\frac{dy}{dx}\) can also be written as \(y'\) or \(\dot{y}\)

⚠️ Common Pitfalls & Tips:

• Order determined by highest derivative present
• General solution needs +C, particular solution has specific C value
• Always check solution by substituting back into original equation
• Initial conditions typically given as y(x₀) = y₀

Separable Differential Equations:

Form: \(\frac{dy}{dx} = f(x)g(y)\)

Can separate variables so all y terms on one side, all x terms on other

Method - Separation of Variables:

1. Write equation as \(\frac{dy}{dx} = f(x)g(y)\)
2. Separate: \(\frac{1}{g(y)}dy = f(x)dx\)
3. Integrate both sides: \(\int \frac{1}{g(y)}dy = \int f(x)dx\)
4. Add constant of integration +C
5. Solve for y if possible (may need to leave implicit)
6. Apply initial condition to find C

Common Models:

1. Exponential Growth/Decay:

\[ \frac{dy}{dx} = ky \]

Solution: \(y = Ae^{kx}\)

k > 0: growth, k < 0: decay

2. Newton's Law of Cooling:

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

T = temperature, T_s = surrounding temperature

3. Logistic Growth:

\[ \frac{dP}{dt} = kP\left(1 - \frac{P}{L}\right) \]

P = population, L = carrying capacity

⚠️ Common Pitfalls & Tips:

• Don't forget to separate variables completely before integrating
• Always add +C after integration
• Check if equation is separable before attempting this method
• May need to use ln rules when integrating 1/y

Solution:

Step 1: Separate variables

\[ \frac{1}{y}dy = x\,dx \]

Step 2: Integrate both sides

\[ \int \frac{1}{y}dy = \int x\,dx \]

\[ \ln|y| = \frac{x^2}{2} + C \]

Step 3: Solve for y

\[ |y| = e^{\frac{x^2}{2} + C} = e^C \cdot e^{\frac{x^2}{2}} \]

\[ y = Ae^{\frac{x^2}{2}} \]

where \(A = \pm e^C\) is an arbitrary constant

Step 4: Apply initial condition y(0) = 2

\[ 2 = Ae^0 = A \]

Therefore A = 2

Answer: \(y = 2e^{\frac{x^2}{2}}\)

Definition:

A slope field is a visual representation of a differential equation

Shows small line segments with slope given by \(\frac{dy}{dx}\) at each point (x, y)

Purpose:

• Visualize behavior of solutions without solving analytically
• Identify equilibrium solutions (horizontal lines)
• See how solutions behave for different initial conditions
• Sketch solution curves through any point

How to Draw Slope Field:

1. For differential equation \(\frac{dy}{dx} = f(x, y)\)
2. Create grid of (x, y) points
3. At each point, calculate slope = f(x, y)
4. Draw short line segment with that slope
5. Solution curves follow the flow of line segments

Reading Slope Fields:

• Solution curves are tangent to slope segments at every point
• Horizontal segments indicate \(\frac{dy}{dx} = 0\) (equilibrium)
• Vertical segments indicate undefined slope
• Solution curves cannot cross each other

⚠️ Common Pitfalls & Tips:

• Use GDC to generate and visualize slope fields
• Solution curves follow the "flow" of the field
• Equilibrium solutions are where all slopes are zero
• Different initial conditions give different solution curves

Definition:

Numerical method to approximate solutions to differential equations

Uses tangent line approximations to step forward

Euler's Method Formula:

For \(\frac{dy}{dx} = f(x, y)\) with initial condition y(x₀) = y₀:

\[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \]

where:

• h = step size (interval width)
• \(x_{n+1} = x_n + h\)
• \(y_n\) = approximate y-value at \(x_n\)

Algorithm:

1. Start with initial point \((x_0, y_0)\)
2. Calculate slope: \(m = f(x_0, y_0)\)
3. Move forward: \(x_1 = x_0 + h\)
4. Estimate: \(y_1 = y_0 + h \cdot m\)
5. Repeat from new point \((x_1, y_1)\)
6. Continue until desired x-value reached

Accuracy:

• Smaller h → better approximation (but more steps)
• Error accumulates with each step
• Best for short intervals

⚠️ Common Pitfalls & Tips:

• Smaller step size h gives better accuracy
• Always start from the initial condition
• Show each iteration step by step in exams
• Use GDC to perform calculations efficiently

Solution:

Given: \(\frac{dy}{dx} = x + y\), h = 0.5, \((x_0, y_0) = (0, 1)\)

Step 1: From x = 0 to x = 0.5

Slope at (0, 1): \(f(0, 1) = 0 + 1 = 1\)

\[ y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.5(1) = 1.5 \]

Point: (0.5, 1.5)

Step 2: From x = 0.5 to x = 1.0

Slope at (0.5, 1.5): \(f(0.5, 1.5) = 0.5 + 1.5 = 2\)

\[ y_2 = y_1 + h \cdot f(x_1, y_1) = 1.5 + 0.5(2) = 2.5 \]

Point: (1.0, 2.5)

Step 3: From x = 1.0 to x = 1.5

Slope at (1.0, 2.5): \(f(1.0, 2.5) = 1.0 + 2.5 = 3.5\)

\[ y_3 = y_2 + h \cdot f(x_2, y_2) = 2.5 + 0.5(3.5) = 4.25 \]

Answer: y(1.5) ≈ 4.25

Coupled Differential Equations:

System where multiple variables depend on each other

\[ \frac{dx}{dt} = f(x, y) \]

\[ \frac{dy}{dt} = g(x, y) \]

Phase Portrait:

Graphical representation in the x-y plane (phase plane)

Shows trajectories (solution curves) of the system

Key Features:

• Equilibrium points: Where \(\frac{dx}{dt} = 0\) and \(\frac{dy}{dt} = 0\)
• Trajectories: Paths showing how system evolves over time
• Direction arrows: Show direction of motion along trajectories

Types of Equilibrium Points:

• Stable (sink): Trajectories converge to point
• Unstable (source): Trajectories diverge from point
• Saddle point: Stable in one direction, unstable in another
• Center: Closed trajectories (periodic solutions)

Predator-Prey Model (Lotka-Volterra):

Classic example of coupled system:

\[ \frac{dR}{dt} = aR - bRP \quad \text{(prey)} \]

\[ \frac{dP}{dt} = -cP + dRP \quad \text{(predator)} \]

R = prey population, P = predator population

⚠️ Common Pitfalls & Tips:

• Equilibrium points are where both derivatives equal zero
• Use GDC to generate phase portraits
• Arrows show direction of motion in time
• Stability determined by behavior near equilibrium

Solving Methods

• Separation: Split x and y
• Integrate both sides
• Apply initial conditions

Slope Fields

• Visual representation
• Solution curves follow flow
• Use GDC to generate

Euler's Method

• \(y_{n+1} = y_n + h \cdot f(x_n, y_n)\)
• Smaller h = better
• Show all steps

Phase Portraits

• Coupled systems
• Equilibrium points
• Trajectory behavior