Differential equations (HL emphasis) - Mathematics: Analysis & Approaches IB Study Notes

Differential equations are equations containing derivatives that describe relationships between functions and their rates of change. A first-order differential equation involves only first derivatives (dy/dx), while second-order equations include d²y/dx².

Key Classifications:

Separable differential equations take the form dy/dx = f(x)g(y), allowing variables to be separated: ∫(1/g(y))dy = ∫f(x)dx. After integration, apply the initial condition (boundary condition) to find the particular solution.

Homogeneous differential equations can be written as dy/dx = f(y/x). Use the substitution v = y/x, so y = vx and dy/dx = v + x(dv/dx), transforming it into a separable equation.

Linear first-order differential equations follow dy/dx + P(x)y = Q(x). The integrating factor method uses μ(x) = e^(∫P(x)dx), multiplying through to create d/dx[μ(x)y] = μ(x)Q(x).

Second-order homogeneous linear equations with constant coefficients: ay'' + by' + cy = 0. Solve using the auxiliary equation am² + bm + c = 0:

• Two distinct real roots m₁, m₂: y = Ae^(m₁x) + Be^(m²x)
• Repeated root m: y = (A + Bx)e^(mx)
• Complex roots α ± βi: y = e^(αx)(A cos βx + B sin βx)

Particular integrals for non-homogeneous equations (ay'' + by' + cy = f(x)) require the method of undetermined coefficients or variation of parameters.

Memory Aid (SHILL): Separable, Homogeneous, Integrating factor, Linear second-order, Linear first-order — the five main types you'll encounter in IB HL exams.

Differential equations model dynamic processes where change depends on current state — think of them as mathematical recipes for change.

Population Growth (Separable): The equation dP/dt = kP describes exponential growth where population change is proportional to current population. Separating: ∫(1/P)dP = ∫k dt gives ln|P| = kt + C, so P = Ae^(kt). With initial condition P(0) = P₀, we get P = P₀e^(kt). This models bacteria colonies, nuclear decay (negative k), and compound interest.

Newton's Law of Cooling (First-Order Linear): dT/dt = -k(T - Tₐ) where T is object temperature and Tₐ is ambient temperature. Rearranging: dT/dt + kT = kTₐ. Using integrating factor e^(kt) yields T = Tₐ + (T₀ - Tₐ)e^(-kt). This applies to forensic science (time of death), coffee cooling, and criminal investigations.

Simple Harmonic Motion (Second-Order): A mass on a spring follows mx'' + kx = 0 (where x is displacement). The auxiliary equation m·r² + k = 0 gives complex roots, producing x = A cos(ωt) + B sin(ωt) where ω = √(k/m). This describes pendulums, electrical circuits (LC oscillators), and molecular vibrations.

Damped Oscillation: Adding friction/resistance: mx'' + cx' + kx = 0. The discriminant b² - 4ac determines:

• Overdamped (distinct real roots): no oscillation
• Critically damped (repeated root): fastest return without oscillation
• Underdamped (complex roots): oscillation with decreasing amplitude

Think of pushing a child on a swing (driving force) with air resistance (damping) — the complete picture requires non-homogeneous equations with particular integrals.