Introduction to Differential Equations

A differential equation is an equation that relates a function to one or more of its derivatives. Rather than asking 'what number satisfies this equation?', a differential equation asks 'what function satisfies this equation?' For example, if you know that a quantity changes at a rate proportional to its current size — like a bacterial colony doubling repeatedly — a differential equation captures that relationship precisely and allows you to find the exact function describing the population at any time. Differential equations appear throughout science and engineering because the natural world is described by rates of change. Newton's second law F = ma is a differential equation in disguise: acceleration is the second derivative of position, so specifying forces on an object gives an equation involving the unknown position function and its derivatives. Similarly, radioactive decay, the cooling of a hot drink, the oscillation of a pendulum, and the charging of a capacitor are all modelled by differential equations. This article introduces the fundamental vocabulary and ideas: what a differential equation is, how to classify equations by order and linearity, what a solution means, and how initial conditions select a unique solution from a whole family. Mastering this language is the essential first step before learning solution techniques.

Key takeaways
  • Definition
  • Order
  • Linearity
  • Solutions and the General Solution
  • Initial Conditions and Particular Solutions
  • Verifying a Solution

Definition

A differential equation (DE) is any equation containing an unknown function and at least one of its derivatives. If the unknown function depends on a single variable (e.g. y(t) or y(x)), the equation is an ordinary differential equation (ODE). If it depends on two or more variables and involves partial derivatives, it is a partial differential equation (PDE). At high-school level we focus exclusively on ODEs.

Order

The order of a differential equation is the highest derivative that appears in it. The equation dy/dx = 3x is first-order because the highest derivative is the first. The equation d²y/dx² + 4y = 0 is second-order. Order tells you how many times you must integrate (roughly speaking) to find the solution, and how many constants of integration — and therefore how many initial or boundary conditions — are needed.

Linearity

A differential equation is linear if the unknown function y and all its derivatives appear only to the first power and are not multiplied together. For example, dy/dx + 2y = e^x is linear, but dy/dx = y² is nonlinear because y is squared. Linear ODEs have a rich solution theory and many can be solved with systematic algebraic methods. Nonlinear ODEs are generally harder; some have no closed-form solution.

Solutions and the General Solution

A solution to a DE is any function y(x) (or y(t)) that satisfies the equation identically — meaning when you substitute it in, both sides are equal for all values of the variable. An n-th order ODE typically has a general solution containing n arbitrary constants. For instance, the first-order ODE dy/dx = 2x has general solution y = x² + C, where C can be any real number. Each value of C gives a different particular solution, and together they form a family of curves.

Initial Conditions and Particular Solutions

An initial condition is an extra piece of information specifying the value of the function (and possibly its derivatives) at a particular point. For example, y(0) = 3 added to dy/dx = 2x forces C = 3, giving the unique particular solution y = x² + 3. An ODE together with initial conditions is called an initial value problem (IVP). Geometrically, an initial condition picks one curve from the family of solution curves — the unique curve passing through the specified point.

Verifying a Solution

To verify that a proposed function is a solution, differentiate it as many times as the order of the ODE, substitute into the equation, and check that both sides are equal. For example, to verify y = e^(3x) is a solution of dy/dx = 3y: differentiate to get dy/dx = 3e^(3x), then check 3e^(3x) = 3·e^(3x). ✓ This substitution approach works for any DE and requires no special technique beyond careful differentiation.

The Key Equation

dydx=ky\frac{dy}{dx} = ky
  • y — Dependent variable (varies (e.g. population, mass, charge)): The unknown function we are trying to find; its value depends on x.
  • x — Independent variable (varies (e.g. time in seconds, distance in metres)): The variable with respect to which we differentiate; often time t.
  • \frac{dy}{dx} — First derivative of y (unit of y per unit of x): The instantaneous rate of change of y; this is what makes the equation a differential equation.
  • k — Growth/decay constant (per unit of x (e.g. s⁻¹)): A constant parameter. If k > 0 the quantity grows exponentially; if k < 0 it decays exponentially. The general solution is y = Ce^(kx).

Worked Example

Problem. Solve the initial value problem: dy/dx = 2x − 1, with y(0) = 4. Then verify your solution.

y=x2−x+4y = x^2 - x + 4

Answer. y = x² − x + 4

Figure
Diagram
Family of Solutions: dy/dx = 2x − 1, i.e. y = x² − x + CxyC=0C=2C=4 (particular)(0, 4)460−111.5Each curve is a solution; the initial condition y(0)=4 selects the red particular solution.
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
GW
Gottfried Wilhelm Leibniz
1646–1716 · German

Leibniz independently developed calculus alongside Newton and introduced the notation dy/dx and ∫ that we still use today. His clear symbolic language made differential equations far easier to manipulate algebraically. He solved several early ODEs, including the separable equation, and corresponded extensively with the Bernoulli brothers who pioneered many solution methods. Leibniz designed a mechanical calculator capable of multiplication and division — the 'Stepped Reckoner' — decades before the computer age, demonstrating his passion for systematising calculation of all kinds.

  • Derivatives and Differentiation — Differential equations are built from derivatives; you must be fluent in differentiating functions before you can verify or solve DEs.
  • Antiderivatives and Indefinite Integrals — Solving the simplest DEs means integrating both sides; the constant of integration C is precisely where the family of solutions comes from.
  • Separable Differential Equations — Separable equations are the next step: a systematic technique for solving first-order ODEs where variables can be separated to opposite sides before integrating.
  • Exponential Growth and Decay — The ODE dy/dt = ky is the most important first-order ODE; its solution y = Ce^(kt) models population, radioactive decay, and compound interest.
  • Slope Fields and Euler's Method — Slope fields provide a graphical way to visualise solution families of ODEs without finding an algebraic formula, connecting geometry to the analytic idea of a derivative.