Differential equation

A differential equation is an equation that relates an unknown function to its own derivatives.

Two features are common to all differential equations:

• the unknown is a function, not a number;
• a solution is rarely unique, but instead often a family of functions containing arbitrary constants. A particular member of the solution family can be singled out by extra conditions, such as the state of the system at some time;

Types of differential equations

Ordinary and partial differential equations

Ordinary differential equations

An ordinary differential equation (ODE) describes an unknown function which depends on a single independent variable.

Writing the unknown as y(x), an ODE can be expressed in its implicit form:

F\bigl(x,y,y',y'',\ldots,y^{(n)}\bigr)=0

where F is a given function.

If the equation can be solved for the highest derivative, it takes an explicit form.

For a first-order equation:

\frac{dy}{dx}=f(x,y)

The right-hand side f(x, y) prescribes the slope of any solution function must have at point (x, y).

Partial differential equations

If the unknown function depends on two or more variables and partial derivatives appear, the equation is a partial differential equation (PDE).

Order and linearity

The order' of a differential equation is the order of the highest derivative that appears in it: an equation containing y⁽ⁿ⁾ but no higher derivative has order n. The special case is when n=1 AND there is no multiplication of two different orders of derivatives (i.e., no term containing yy), where the differential equation is said to be linear.

Differential equations as mathematical models

Mechanics

Force and acceleration

Newton's second law of motion states that the acceleration of a body is proportional to the net force acting on it:

m\frac{d^{2}x}{dt^{2}}=F

where x(t) is the position of the body, m its mass, and the force F may itself depend on time, position and velocity.

Harmonic oscillations

For a mass attached to a spring whose restoring force is linear in the displacement, this becomes the harmonic oscillator equation:

\frac{d^{2}x}{dt^{2}}+\omega_{0}^{2}x=0

whose solutions are sinusoidal oscillations at the natural frequency ω₀; such motion is called simple harmonic motion.

A mass on a spring executing simple harmonic motion, the solution of the harmonic oscillator equation. Credit: Evil saltine (public domain).

Exponential growth and decay

A quantity that changes at a rate proportional to its current size satisfies the exponential growth or decay equation:

\frac{dy}{dt}=k\,y

Its solutions are exponentials:

y(t)=y_{0}\,e^{kt}

When k > 0, it can describe a population with unlimited resources that grows exponentially, whereas k < 0 describes a radioactive substance that decays exponentially towards zero.

Heat transfer

Newton's law of cooling describes a body whose temperature T differs from a constant ambient temperature Tₐ:

\frac{dT}{dt}=-k\bigl(T-T_{a}\bigr)

The temperature difference decays exponentially, so the body approaches the ambient temperature but never quite reaches it.

Geometry of first-order equations

Direction fields

As described above, a first-order equation has the explicit form dy/dx = f(x, y)

which assigns to every point (x, y) the slope that a solution curve must have there.

Drawing a short line segment of that slope at many points produces a direction field (or slope field). Every solution curve must stay tangent to the field directions everywhere.

Direction field of the equation dy/dx = y, with several solution curves. Each segment shows the local slope; the solutions are exponential curves. Credit: jjbeard (public domain).

As such, direction fields make the qualitative behaviour of solutions visible even when no closed-form solution exists.

Solving differential equations

Analytical methods

For the relatively small class of equations that can be solved in closed form, a toolkit of classical methods exists:

• first-order equations: separation of variables, integrating factors for linear equations, substitutions such as the Bernoulli substitution;
• linear equations with constant coefficients: exponential trial solutions and the characteristic equation, variation of parameters;
• equations with variable coefficients: power-series solutions about ordinary points and Frobenius series about regular singular points;
• integral transforms, in particular the Laplace transform, which turns differential equations into algebraic ones;
• for partial differential equations: separation of variables combined with Fourier series, and transform methods.

The theory of these methods is treated in standard textbooks on ordinary and partial differential equations. Still, only a minority of differential equations, almost all of them linear, can be solved explicitly; most equations that arise in applications are treated by numerical or qualitative methods.

Numerical methods

When no closed-form solution is available, solutions are approximated by numerical integration. The simplest scheme, Euler's method, advances an approximate solution yₙ at points xₙ = x₀ + nh by repeatedly applying one step:

y_{n+1}=y_{n}+h\,f(x_{n},y_{n})

More accurate and more stable methods, such as the Runge–Kutta family, are built on the same idea of using the differential equation to extrapolate from known values, and are implemented in essentially every numerical computing environment.

Qualitative methods

For nonlinear equations, and especially for systems of two or more equations, the geometry of solutions matters as much as their formulas. Qualitative analysis studies the equilibrium points at which the system does not change, the stability of those equilibria, and the way solutions flow between them in the phase plane; as parameters change, the behaviour can reorganise itself in bifurcations, and in three or more dimensions a system can settle onto a chaotic attractor, whose solutions are deterministic yet aperiodic and extremely sensitive to initial conditions.

History

Portrait of Isaac Newton (after Godfrey Kneller, 1689). Newton's laws of motion and of universal gravitation, published in the Principia (1687), are differential equations. Credit: James Thronill after Godfrey Kneller (public domain).

Differential equations arose with the calculus that Isaac Newton and Gottfried Wilhelm Leibniz developed independently in the late 17th century. Newton's laws of motion and of universal gravitation, published in his Principia of 1687, are differential equations. However, it was Leibniz who introduced the notation dy/dx still used today.

In the 18th century Leonhard Euler turned a collection of ad hoc tricks into a systematic theory, developing the general treatment of linear equations with constant coefficients, series methods, and the first numerical integration scheme, which still bears his name.

Leonhard Euler (portrait by Jakob Emanuel Handmann, 1753). Euler created much of the systematic theory of differential equations and gave his name to Euler's method. Credit: Jakob Emanuel Handmann (public domain).

Alongside the theory of ordinary equations, the 18th and 19th centuries saw the development of partial differential equations of physics. Jean le Rond d'Alembert wrote down and solved the one-dimensional wave equation for the vibrating string in the 1740s, and Joseph Fourier derived the heat equation from the physics of conduction and solved it with trigonometric series in his Théorie analytique de la chaleur of 1822, founding Fourier analysis. Augustin-Louis Cauchy and, later, Rudolf Lipschitz and Émile Picard placed the existence and uniqueness of solutions on a rigorous footing.

The 20th century saw the development of the modern theory of dynamical systems. In 1963, meteorologist Edward Lorenz found chaotic behaviour in a simple system of three differential equations modelling atmospheric convection, a landmark in the study of nonlinear dynamics.

Further reading

• Differential equation, Wikipedia
• Differential Equation, Wolfram MathWorld
• 18.03SC Differential Equations, MIT OpenCourseWare
• Leonhard Euler, MacTutor History of Mathematics