HL: differential equations and advanced functions - Mathematics: Analysis & Approaches IB Study Notes

Differential equations are equations involving derivatives that describe how quantities change. In HL Mathematics: Analysis & Approaches, you'll encounter first-order differential equations (involving dy/dx) and second-order differential equations (involving d²y/dx²).

Key types of first-order differential equations:

1. Separable equations: Form dy/dx = f(x)g(y), solved by separating variables: ∫(1/g(y))dy = ∫f(x)dx

2. Linear first-order: Form dy/dx + P(x)y = Q(x), solved using integrating factor μ(x) = e^(∫P(x)dx)

3. Homogeneous equations: Form dy/dx = f(y/x), solved by substitution v = y/x

Second-order differential equations include:

• Homogeneous linear: a(d²y/dx²) + b(dy/dx) + cy = 0
• Non-homogeneous linear: a(d²y/dx²) + b(dy/dx) + cy = f(x)

For homogeneous equations, find the auxiliary equation am² + bm + c = 0. Solutions depend on the discriminant Δ = b² - 4ac:

• Δ > 0: Two real roots m₁, m₂ → y = Ae^(m₁x) + Be^(m₂x)
• Δ = 0: Repeated root m → y = (A + Bx)e^(mx)
• Δ < 0: Complex roots α ± βi → y = e^(αx)(A cos βx + B sin βx)

Advanced functions include inverse functions f⁻¹(x), composite functions (f ∘ g)(x), piecewise functions, and implicit functions. The domain and range must be carefully identified, especially when restricting functions to make them one-to-one for invertibility.

Memory aid (SAIL): Separable, Auxiliary equation, Integrating factor, Linear forms

Differential equations are the mathematical language of change — they model everything from population growth to radioactive decay to electrical circuits.

Population dynamics: The equation dP/dt = kP describes exponential growth, where population P grows proportionally to its current size. Solution: P = P₀e^(kt). When resources are limited, we use the logistic equation dP/dt = kP(1 - P/M), where M is carrying capacity. This models how bacterial colonies grow rapidly initially, then slow as nutrients deplete.

Newton's Law of Cooling: The temperature T of an object cooling in ambient temperature T_a follows dT/dt = -k(T - T_a). This separable equation explains why your coffee cools quickly at first (large temperature difference) then slowly approaches room temperature. Solution: T = T_a + (T₀ - T_a)e^(-kt).

Mechanical oscillations: A mass on a spring follows d²x/dt² + (k/m)x = 0, producing simple harmonic motion x = A cos(ωt) + B sin(ωt). Adding damping (friction) creates d²x/dt² + c(dx/dt) + (k/m)x = 0, modeling everything from car suspensions to earthquakes.

Electric circuits: An RC circuit follows the equation Q/C + R(dQ/dt) = V(t), where Q is charge. This first-order linear equation describes charging/discharging capacitors in smartphones and computers.

Think of differential equations as recipes: Instead of telling you the final cake (the function), they describe how ingredients combine and transform (the derivatives). Your job is to reverse-engineer the original function through integration and clever manipulation.

Analogy: Separable equations are like unmixing ingredients — you isolate x-terms on one side, y-terms on the other, then integrate separately to "undo" the mixing.