differential equations

Differential equations are equations involving derivatives that describe how quantities change. In Cambridge A-Level, you'll encounter first-order differential equations (containing dy/dx but not higher derivatives) and simple second-order equations.

Key Types of First-Order Differential Equations:

1. Variables Separable: dy/dx = f(x)g(y), solved by rearranging to ∫1/g(y) dy = ∫f(x) dx
2. Integrating Factor Method: For dy/dx + P(x)y = Q(x), multiply through by integrating factor μ(x) = e^(∫P(x)dx)

General vs Particular Solutions: The general solution contains an arbitrary constant C (representing a family of curves). A particular solution satisfies specific initial conditions or boundary conditions (e.g., y = 3 when x = 0).

Essential Formulas:

• Separable form: ∫h(y) dy = ∫g(x) dx + C
• Integrating factor: μ(x) = e^(∫P(x)dx), then d/dx[μy] = μQ(x)

Second-Order Equations (limited scope): For a" + ba' + c = 0, the auxiliary equation is m² + bm + c = 0. Real distinct roots give y = Ae^(m₁x) + Be^(m₂x); repeated roots give y = (A + Bx)e^(mx); complex roots m = α ± βi give y = e^(αx)(A cos βx + B sin βx).

Mnemonic: DISC - Determine type, Isolate variables, Solve integrals, Check solution by differentiation.

Cambridge Command Words: Solve means find general solution; find the particular solution requires using given conditions; verify means substitute back to confirm.

Differential equations model rates of change in countless real-world scenarios, making them powerful tools in science, engineering, and economics.

Population Growth (Variables Separable): Consider bacteria population P growing at a rate proportional to current population: dP/dt = kP. Separating variables: ∫1/P dP = ∫k dt gives ln|P| = kt + C, so P = Ae^(kt). This is exponential growth—think of viral spread during pandemics or compound interest in banking. The constant A represents initial population.

Newton's Law of Cooling (Integrating Factor): A cup of coffee cools according to dT/dt = -k(T - Tᵣₒₒₘ), where T is coffee temperature. This rearranges to dT/dt + kT = kTᵣₒₒₘ, solved using integrating factor e^(kt). The solution shows temperature approaches room temperature exponentially—explaining why your coffee cools quickly at first, then slowly.

Simple Harmonic Motion (Second-Order): A mass on a spring follows d²x/dt² = -ω²x. The solution x = A cos(ωt) + B sin(ωt) describes oscillation—from pendulums to atoms vibrating in molecules.

Analogy: Think of differential equations as detective work. You're given clues about how something changes (the derivative), and you must reconstruct the complete story (the original function). Initial conditions are like witnesses narrowing down possibilities from many suspects (general solution) to the culprit (particular solution).

Real Application: Engineers use dv/dt = g - kv/m to model parachute descent, determining safe landing velocities by finding terminal velocity where dv/dt = 0.