IB Mathematics AA – Topic 5: Calculus

Comprehensive Guide to Differential Equations

Introduction to Differential Equations

Differential equations are equations involving unknown functions and their derivatives, providing the mathematical language for describing dynamic systems where rates of change matter. From population growth and radioactive decay to cooling temperatures and chemical reactions, differential equations model how quantities change over time based on their current state and the forces acting upon them.

Key concepts: A first-order differential equation involves the first derivative of a function. Solving these equations means finding the function itself—not just its derivative. Different types require different solution techniques: separable equations allow us to isolate variables, homogeneous equations use clever substitutions, and linear equations employ integrating factors. When analytical solutions are impossible or impractical, Euler's method provides numerical approximations.

Why differential equations matter: Differential equations are fundamental to modeling the real world. Engineers use them to design control systems, economists to model market dynamics, biologists to predict population changes, physicists to describe motion and forces, and medical researchers to understand disease spread. The ability to set up, solve, and interpret differential equations transforms theoretical mathematics into practical problem-solving tools across all scientific disciplines.

In this guide: We'll master first-order differential equations and their classification, apply Euler's method for numerical solutions, solve separable differential equations by isolating variables, tackle homogeneous equations using substitution techniques, apply the integrating factor method for linear equations, and solve real-world applications including exponential growth, decay, and Newton's law of cooling—all essential skills for IB HL exam success.

1. First-Order Differential Equations

Definition and Classification

First-Order Differential Equation

\(\frac{dy}{dx} = f(x, y)\)

Relates first derivative to function of \(x\) and \(y\)

Key Terminology:

Order: Highest derivative present (first-order has \(\frac{dy}{dx}\))
General solution: Contains arbitrary constant \(C\)
Particular solution: Specific solution using initial conditions
Initial condition: Given value like \(y(0) = 2\) to find \(C\)

Types of First-Order Equations

Classification:

Type	Form	Method
Separable	\(\frac{dy}{dx} = g(x)h(y)\)	Separate and integrate
Homogeneous	\(\frac{dy}{dx} = F(\frac{y}{x})\)	Substitute \(v = \frac{y}{x}\)
Linear	\(\frac{dy}{dx} + P(x)y = Q(x)\)	Integrating factor

2. Euler's Method (Numerical Solution)

The Euler Formula

Euler's Method

Given \(\frac{dy}{dx} = f(x,y)\) with initial condition \((x_0, y_0)\):

\(x_{n+1} = x_n + h\)

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

where \(h\) is the step size

Step-by-Step Process

Euler's Method Steps:

Start with initial values \(x_0\) and \(y_0\)
Choose step size \(h\) (usually given in exam)
Calculate next \(x\): \(x_{n+1} = x_n + h\)
Evaluate \(f(x_n, y_n)\) using the differential equation
Calculate next \(y\): \(y_{n+1} = y_n + h \cdot f(x_n, y_n)\)
Repeat for required number of steps

💡 Euler's Method Tips:

Create a table with columns: \(n\), \(x_n\), \(y_n\), \(f(x_n, y_n)\), \(y_{n+1}\)
Smaller step size \(h\) gives more accurate approximation
Euler gives approximate solutions—not exact!
Always show working row by row in exam

Example 1: Euler's Method

Problem: Use Euler's method with step size \(h = 0.2\) to approximate \(y(0.6)\) for \(\frac{dy}{dx} = x + y\) with initial condition \(y(0) = 1\)

Solution:

Setup:

\(x_0 = 0\), \(y_0 = 1\), \(h = 0.2\)

\(f(x,y) = x + y\)

Need 3 steps to reach \(x = 0.6\)

Step 1: \(n = 0\)

\(x_0 = 0\), \(y_0 = 1\)

\(f(0, 1) = 0 + 1 = 1\)

\(y_1 = y_0 + h \cdot f(x_0, y_0) = 1 + 0.2(1) = 1.2\)

\(x_1 = 0 + 0.2 = 0.2\)

Step 2: \(n = 1\)

\(x_1 = 0.2\), \(y_1 = 1.2\)

\(f(0.2, 1.2) = 0.2 + 1.2 = 1.4\)

\(y_2 = 1.2 + 0.2(1.4) = 1.2 + 0.28 = 1.48\)

\(x_2 = 0.2 + 0.2 = 0.4\)

Step 3: \(n = 2\)

\(x_2 = 0.4\), \(y_2 = 1.48\)

\(f(0.4, 1.48) = 0.4 + 1.48 = 1.88\)

\(y_3 = 1.48 + 0.2(1.88) = 1.48 + 0.376 = 1.856\)

\(x_3 = 0.4 + 0.2 = 0.6\)

\(y(0.6) \approx 1.856\) (or 1.86 to 3 s.f.)

3. Separable Differential Equations

Definition and Method

Separable Form

\(\frac{dy}{dx} = g(x) \cdot h(y)\)

Can separate variables: all \(y\)'s on one side, all \(x\)'s on other

Solution Method:

Separate variables: \(\frac{1}{h(y)} dy = g(x) dx\)
Integrate both sides: \(\int \frac{1}{h(y)} dy = \int g(x) dx\)
Add constant of integration \(+C\) on one side only
Solve for \(y\) if possible
Use initial conditions to find \(C\)

⚠ Separable Equation Pitfalls:

Division check: When dividing by \(h(y)\), note if \(h(y) = 0\) gives additional solutions
One constant only: Add \(+C\) to one side only (usually right side)
Integrate carefully: Don't forget absolute value for \(\ln\) functions
Explicit form: Try to solve for \(y =\) explicitly when possible

Example 2: Separable Equation (IB-Style)

Problem: Solve \(\frac{dy}{dx} = \frac{2xy}{1+x^2}\) with initial condition \(y(0) = 3\)

Solution:

Step 1: Verify separable

Equation is \(\frac{dy}{dx} = \frac{2x}{1+x^2} \cdot y\)

Form is \(g(x) \cdot h(y)\) where \(g(x) = \frac{2x}{1+x^2}\) and \(h(y) = y\)

✓ Separable

Step 2: Separate variables

\(\frac{1}{y} dy = \frac{2x}{1+x^2} dx\)

Step 3: Integrate both sides

\(\int \frac{1}{y} dy = \int \frac{2x}{1+x^2} dx\)

Left side: \(\ln|y|\)

Right side: Let \(u = 1+x^2\), \(du = 2x dx\)

\(\int \frac{1}{u} du = \ln|u| = \ln|1+x^2|\)

\(\ln|y| = \ln(1+x^2) + C\)

Step 4: Solve for \(y\)

\(|y| = e^{\ln(1+x^2) + C} = e^C \cdot (1+x^2)\)

Let \(A = e^C\) (positive constant)

\(y = A(1+x^2)\) (taking positive solution)

Step 5: Apply initial condition

\(y(0) = 3\)

\(3 = A(1+0) = A\)

\(A = 3\)

Solution: \(y = 3(1+x^2)\)

4. Homogeneous Equations & Integrating Factor

Homogeneous Equations

Homogeneous Form:

\(\frac{dy}{dx} = F\left(\frac{y}{x}\right)\)

Solution Method:

Substitute \(v = \frac{y}{x}\), so \(y = vx\)
Find \(\frac{dy}{dx} = v + x\frac{dv}{dx}\) (product rule)
Substitute into equation: \(v + x\frac{dv}{dx} = F(v)\)
Rearrange: \(x\frac{dv}{dx} = F(v) - v\)
Separate variables and solve for \(v\)
Substitute back: \(y = vx\)

Linear Equations - Integrating Factor Method

Linear First-Order DE

\(\frac{dy}{dx} + P(x)y = Q(x)\)

Integrating Factor:

\(\mu(x) = e^{\int P(x) dx}\)

Solution Steps:

Rewrite in standard form \(\frac{dy}{dx} + P(x)y = Q(x)\)
Calculate integrating factor \(\mu(x) = e^{\int P(x) dx}\)
Multiply entire equation by \(\mu(x)\)
Left side becomes \(\frac{d}{dx}[\mu(x)y]\)
Integrate: \(\mu(x)y = \int \mu(x)Q(x) dx + C\)
Solve for \(y\): \(y = \frac{1}{\mu(x)}\left[\int \mu(x)Q(x) dx + C\right]\)

💡 Integrating Factor Tips:

Don't add \(+C\) when finding \(\mu(x)\)—any antiderivative works
Check your work: left side should equal \(\frac{d}{dx}[\mu y]\)
Remember to divide by \(\mu(x)\) at the end
Use initial conditions after finding general solution

5. Applications of Differential Equations

Exponential Growth and Decay

Growth/Decay Model:

\(\frac{dy}{dt} = ky\)

Solution: \(y = y_0 e^{kt}\)

\(k > 0\): Exponential growth (population, compound interest)
\(k < 0\): Exponential decay (radioactive decay, cooling)
\(y_0\): Initial value at \(t = 0\)

Newton's Law of Cooling

Cooling Model:

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

Solution: \(T = T_a + (T_0 - T_a)e^{-kt}\)

\(T\): Temperature at time \(t\)
\(T_a\): Ambient (surrounding) temperature
\(T_0\): Initial temperature
\(k\): Cooling constant (\(k > 0\))

Logistic Growth

Limited Growth Model:

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

\(P\): Population at time \(t\)
\(L\): Carrying capacity (maximum sustainable population)
Growth rate slows as \(P\) approaches \(L\)

📋 Differential Equations Quick Reference

Type	Form	Method
Separable	\(\frac{dy}{dx} = g(x)h(y)\)	Separate and integrate
Homogeneous	\(\frac{dy}{dx} = F(\frac{y}{x})\)	Substitute \(v = \frac{y}{x}\)
Linear	\(\frac{dy}{dx} + Py = Q\)	Integrating factor \(\mu = e^{\int P dx}\)
Numerical	Any \(\frac{dy}{dx} = f(x,y)\)	Euler's method

🎯 IB Exam Strategy

Common Question Types:

"Use Euler's method": Create table, show each iteration clearly
"Solve the differential equation": Identify type first (separable, linear, etc.)
"Given initial condition": Find general solution first, then apply condition
"Model growth/decay": Set up DE, solve, interpret constants
"Find when...": Solve equation for \(t\) after finding solution

Key Reminders:

Always identify the type of DE before solving
For separable: separate completely before integrating
For linear: calculate integrating factor carefully
Euler's method: show all steps in a table
Initial conditions: apply AFTER finding general solution
Check: substitute your solution back into original equation

🎉 Master Differential Equations!

Differential equations are the language of change in the natural world. From modeling population dynamics to understanding cooling processes, from predicting exponential growth to analyzing chemical reactions, differential equations provide the mathematical framework for describing how systems evolve over time. Master solution techniques, numerical methods, and real-world applications to excel in IB HL exams and prepare for advanced mathematics and sciences!

Key Success Factors:

✓ Identify type first: separable, homogeneous, or linear
✓ Euler's method: \(y_{n+1} = y_n + h \cdot f(x_n, y_n)\)
✓ Separable: \(\int \frac{dy}{h(y)} = \int g(x) dx\)
✓ Linear: integrating factor \(\mu = e^{\int P(x) dx}\)
✓ Growth/decay: \(\frac{dy}{dt} = ky\) gives \(y = y_0e^{kt}\)
✓ Always apply initial conditions to find constant \(C\)

Classify Correctly • Apply Methods • Interpret Solutions

Master differential equations and excel in IB Mathematics! 🚀