Differential equation models (HL emphasis) - Mathematics: Applications & Interpretation IB Study Notes

Differential equations are equations involving derivatives that model rates of change in real-world phenomena. In IB Mathematics: Applications & Interpretation HL, you'll work with first-order differential equations in the form dy/dx = f(x,y).

Key Types of Differential Equations:

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

2. First-order linear equations: dy/dx + P(x)y = Q(x), requiring an integrating factor μ(x) = e^(∫P(x)dx)

Fundamental Formulas:

• Exponential growth/decay: dy/dt = ky, solution: y = Ae^(kt)
• Logistic growth: dy/dt = ky(1 - y/L), where L is carrying capacity
• Newton's Law of Cooling: dT/dt = -k(T - T_s), where T_s is surrounding temperature

The separation of variables method works when you can write dy/dx = f(x)/g(y). Rearrange to g(y)dy = f(x)dx, then integrate both sides.

Slope fields (direction fields) provide graphical representations showing solution curves without solving algebraically. Each small line segment has gradient dy/dx at point (x,y).

Memory Aid -SIDE: Separate variables, Integrate both sides, Don't forget +C, Evaluate using initial conditions.

Initial value problems (IVPs) specify y(x₀) = y₀, allowing determination of the arbitrary constant C. This gives a particular solution rather than the general solution containing C.

Understanding the qualitative behavior through equilibrium solutions (where dy/dx = 0) and stability analysis is crucial for modeling applications.

Differential equations are the mathematical language of change, describing everything from population dynamics to medication absorption in the bloodstream.

Population Growth Models:

Imagine bacteria in a petri dish. Exponential growth (dP/dt = kP) assumes unlimited resources—population doubles regularly. Think of it like compound interest: the rate of growth depends on current population size. However, this model breaks down as resources deplete.

Logistic growth adds realism: dP/dt = kP(1 - P/L). Here, L is the carrying capacity—the maximum sustainable population. When P is small, growth is nearly exponential. As P approaches L, growth slows like a car approaching its speed limit. The factor (1 - P/L) acts as a "brake" on growth.

Analogy: Exponential growth is like posting a viral video with no limits; logistic growth is like filling a theater—growth slows as seats fill up.

Newton's Law of Cooling:

A hot coffee cooling follows dT/dt = -k(T - T_room). The rate of cooling is proportional to the temperature difference. When your coffee is very hot, it cools quickly. As it approaches room temperature, cooling slows—you've experienced this differential equation!

Drug Concentration:

Medicine elimination follows dC/dt = -kC (first-order kinetics). If C is drug concentration, the body eliminates a percentage per hour, not a fixed amount. This explains why half-life is constant regardless of initial dose.

Mixing Problems:

A tank with 100L of water receiving salt solution follows dS/dt = (rate in) - (rate out). This models environmental pollution in lakes or chemical reactions in industry—Cambridge loves these contexts for examination questions!