Most of the laws that describe the physical world are not equations relating quantities — they are equations relating quantities and their rates of change. Newton's F=ma says force equals mass times acceleration, which is mass times the second derivative of position. That is a differential equation. The heat equation, the wave equation, Maxwell's equations for electromagnetism — all differential equations.
This page covers the basics: what a differential equation is, how solutions work, and the two types you will encounter in a first course — separable ODEs and first-order linear ODEs.
What is a Differential Equation?
A differential equation (DE) is an equation involving a function and its derivatives. Example: dy/dx = ky (exponential growth). Solution: y=Ce^(kx) — a function, not a number. DEs describe how quantities change: population, temperature, current, position.
Classification
- ODE vs PDE: Ordinary DEs involve one variable; Partial DEs involve partial derivatives of multiple variables.
- Order: the order of the highest derivative (first-order: dy/dx; second-order: d²y/dx²).
- Linear vs Nonlinear: linear if y and its derivatives appear only to the first power with no products.
Separable First-Order DEs
- Form: dy/dx = f(x)·g(y). Method: separate variables: dy/g(y) = f(x)dx, then integrate both sides.
- Example: dy/dx = 2xy. Separate: dy/y = 2x dx. Integrate: ln|y| = x²+C. Solve: y = Ae^(x²).
Linear First-Order DEs
Form: dy/dx + P(x)y = Q(x). Method: multiply by integrating factor μ(x)=e^(∫P dx). Then d/dx[μy] = μQ. Integrate both sides.
Applications — Exponential Growth and Decay
dy/dt = ky → y = y₀e^(kt). k>0: exponential growth (population, compound interest). k<0: exponential decay (radioactive decay, cooling). Half-life T½ = ln(2)/|k|.
A differential equation is an equation relating a function to its derivatives. The solution is a function (not a number). DEs describe how quantities change in time or space — from population growth to planetary motion to heat flow.
Why Differential Equations Are Everywhere
Newton's second law F = ma is a differential equation: force equals mass times the second derivative of position. Maxwell's equations are differential equations — they govern all electromagnetic phenomena including light. The Schrödinger equation in quantum mechanics is a differential equation. The Black-Scholes equation in finance is a differential equation. Epidemiological models (SIR models for disease spread) are systems of differential equations. DEs are not a niche topic — they are the primary language in which science describes how the universe evolves.
Classification of Differential Equations
By type: Ordinary DEs (ODEs) involve one independent variable. Partial DEs (PDEs) involve partial derivatives with respect to multiple variables. By order: The order equals the highest derivative present. y' = ky is first-order. y'' + y = 0 is second-order. By linearity: An ODE is linear if y and all its derivatives appear only to the first power with no products between them. y'' + 3y' + 2y = eˣ is linear. y'' + y² = 0 is nonlinear. Linear ODEs have a much richer and more complete theory than nonlinear ones.
Separable ODEs — Full Method
A separable ODE has the form dy/dx = f(x)g(y). Separate variables: dy/g(y) = f(x)dx. Integrate both sides: ∫dy/g(y) = ∫f(x)dx + C. Then solve for y if possible.
First-Order Linear ODEs — Integrating Factor
Standard form: dy/dx + P(x)y = Q(x). Multiplying by the integrating factor μ(x) = e^(∫P dx) gives d/dx[μy] = μQ, which integrates directly. Procedure: (1) Write in standard form. (2) Compute μ = e^(∫P dx). (3) Multiply through by μ. (4) Integrate both sides: μy = ∫μQ dx + C. (5) Solve for y.
Second-Order Linear ODEs with Constant Coefficients
Form: ay'' + by' + cy = 0. Try y = eˢˣ: substituting gives the characteristic equation as² + bs + c = 0. The nature of solutions depends on the discriminant b² − 4ac: positive → two real roots s₁, s₂ → y = C₁e^(s₁x) + C₂e^(s₂x). Zero → one repeated root s → y = (C₁ + C₂x)e^(sx). Negative → complex roots α ± βi → y = eˢˣ(C₁ cos βx + C₂ sin βx). This last case describes oscillatory behaviour — it is the mathematics of springs, pendulums, and electrical oscillators.
Slope Fields and Qualitative Analysis
A slope field (direction field) visualises a first-order ODE by drawing small line segments of slope f(x,y) at each point (x,y). Without solving the ODE analytically, you can read off: equilibrium solutions (where the slope is always zero — horizontal tangent lines everywhere), stability (whether nearby solutions converge to or diverge from equilibria), and the qualitative shape of all solutions. This geometric approach is powerful when the ODE cannot be solved in closed form — which is the case for most nonlinear ODEs.
- Boyce, W.E. & DiPrima, R.C. (2017). Elementary Differential Equations, 11th ed. Wiley.
- Simmons, G.F. (2016). Differential Equations with Applications. McGraw-Hill.
- Stewart, J. (2015). Calculus, §9.1–9.5. Cengage.