Introduction to Differential Equations

The Language of Change

Differential equations are the language in which nature describes change. When a physicist writes Newton’s second law F = ma, they are essentially writing a differential equation: acceleration a = x'' is the second derivative of position x with respect to time. The heat conduction equation, Maxwell’s equations, the Schrödinger equation of quantum mechanics, the Black–Scholes equation for option pricing—all are differential equations. The study of differential equations is the study of how a changing world is structured.

The history of differential equations is inseparable from the history of physics. Newton created mathematical analysis precisely to solve mechanics problems—and the first differential equations in the history of science were equations describing the motion of planets. Since then, the field has spread to biology (population growth, epidemic spread), chemistry (reaction kinetics), economics (optimal control, growth models), engineering (system control, circuit theory).

Basic Concepts

ODE (ordinary differential equation) links a function of one variable y(x) to its derivatives: F(x, y, y', y'', ..., y^(n)) = 0. The word "ordinary" distinguishes them from partial differential equations, where the function depends on several variables.

The order of an ODE is the highest order of derivative that appears in the equation. The equation y' = ky is first order; y'' + ω²y = 0 (harmonic oscillator) is second order.

The normal form of a first-order ODE: y' = f(x, y). This form is the starting point for most theoretical results and numerical methods.

Cauchy Problem

A single equation y' = f(x, y) has infinitely many solutions—a family of integral curves that fill the domain. To select a specific solution, an initial condition is needed: y(x₀) = y₀.

Cauchy Problem: find a solution to y' = f(x, y) that satisfies y(x₀) = y₀.

Geometrically: at each point (x, y), the function f(x, y) specifies the direction (slope) of the tangent to the integral curve. The set of these directions forms a direction field. The solution to the Cauchy problem is a curve that "flows" along the direction field and passes through the given point (x₀, y₀). Constructing the direction field is the first tool for qualitative analysis of differential equations without their explicit solution.

Equations with Separable Variables

The simplest class of ODEs permitting explicit solutions: y' = f(x)g(y). The variables "separate" onto the left and right parts.

Method: divide both sides by g(y) (assuming g(y) ≠ 0) and integrate:

dy/g(y) = f(x) dx → ∫ dy/g(y) = ∫ f(x) dx + C.

Detailed Example: Solve y' = xy with initial condition y(0) = 2.

Separate: dy/y = x dx. Integrate left side: ∫ dy/y = ln|y|. Right side: ∫ x dx = x²/2. Obtain ln|y| = x²/2 + C, from which y = Ae^(x²/2), where A = e^C. Apply initial condition: y(0) = A·1 = 2, so A = 2. Answer: y = 2e^(x²/2).

Law of Exponential Growth and Decay

Model: N' = kN — the most important equation of applied mathematics.

This equation has separable variables: dN/N = k dt → ln N = kt + C → N(t) = N₀e^(kt).

For k > 0: exponential growth. Applications: population growth (Malthus model), multiplication of bacteria in nutrient medium, interest accrual in a bank. If initially there are 1000 bacteria and they double every 20 minutes (k = ln2/20 min⁻¹), after 2 hours there will be 1000 · 2⁶ = 64,000.

For k < 0: exponential decay. Applications: radioactive decay, cooling of a body (Newton’s law), investment depreciation. Half-life period — time T over which the amount of substance halves: N₀/2 = N₀e^(kT) → T = -ln(2)/k = ln(2)/|k|.

For carbon-14, used in radiocarbon dating: T = 5730 years. If a sample retains 30% of its original amount, its age is: 0.3 = e^(-t/8267) → t = 8267 · ln(1/0.3) ≈ 9950 years.

Homogeneous Equations

The equation y' = f(y/x) is called homogeneous. The substitution v = y/x (i.e., y = vx) reduces it to an equation with separable variables.

From y = vx: y' = v + xv'. Substitute: v + xv' = f(v) → xv' = f(v) − v → dv/(f(v) − v) = dx/x.

Example: y' = (y² + xy)/x² = (y/x)² + (y/x). Substitution v = y/x: v + xv' = v² + v, from which xv' = v², that is dv/v² = dx/x. Integrate: -1/v = ln x + C. Returning to y: -x/y = ln x + C or y = -x/(ln x + C).

Linear First-Order Equations

The equation y' + p(x)y = q(x) is linear because y and y' appear to the first power. The solution method is variation of constants (or integrating factor method).

Step 1: Solve the homogeneous equation y' + p(x)y = 0: y₀ = Ce^(−∫p dx).

Step 2: Seek a solution in the form y = C(x)e^(−∫p dx). Substitute into the equation—C(x) is found by integrating.

Final formula: y = e^(−∫p dx)[∫q(x)e^(∫p dx) dx + C].

Example: y' − y = eˣ. Here p = -1, q = eˣ. Integrating factor μ = e^(-x). The equation becomes: (ye^(-x))' = 1. Integrate: ye^(-x) = x + C. Answer: y = (x + C)eˣ.

Linear first-order ODEs model RL-circuits in electrical engineering (current as a function of time), mixing problems (concentration of a substance in a tank), heat outflow problems in thermodynamics.

Question for reflection: The Malthusian growth model N' = kN predicts unlimited growth. Why do real populations not grow indefinitely? What equation models limited growth better?