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 that any solution function must have at the point (x, y).

For an equation of order n that is solved for its highest derivative:

y^{(n)}=f\bigl(x,y,y',y'',\ldots,y^{(n-1)}\bigr)

Integrating the equation once removes one derivative and introduces one arbitrary constant. Therefore, a problem of order n generally needs n conditions to single out one particular solution of the solution family.

Partial differential equations

A partial differential equation describes an unknown function that depends on two or more independent variables. Its general form is:

F\Bigl(x_1,x_2,\ldots,x_n,\ u,\ \frac{\partial u}{\partial x_1},\ldots,\ \frac{\partial^{2}u}{\partial x_i\,\partial x_j},\ldots\Bigr)=0

Order and linearity

The order of a differential equation is the order of the highest derivative that appears in it.

A differential equation is linear if the unknown function and its derivatives appear only to the first power and are never multiplied together (they may be multiplied by functions of the independent variables).

An n-th order linear ODE therefore has the form

a_n(x)\,y^{(n)}+a_{n-1}(x)\,y^{(n-1)}+\cdots+a_1(x)\,y'+a_0(x)\,y=g(x)

We consider this equation homogeneous when g(x) = 0.

Linear homogeneous equations obey the superposition principle: any linear combination of solutions is again a solution.

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 can be written explicitly

\frac{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 can visualise the qualitative behaviour of solutions.

Solving differential equations

Solutions to differential equations can be classified into 3 categories:

• Closed-form solution: a finite combination of elementary functions (polynomials, rational functions, exponentials, logarithms, trigonometric functions and their inverses) built with the usual algebraic operations, which can be evaluated directly at a given point. Only a small fraction of differential equations admit a closed-form solution.

• Open form solution: a combination of elementary functions, and/or infinite series (a power series or a Fourier series) or special functions defined by such series, and integrals.

• numerical estimation: approximation of the real solution to a desired accuracy.

Analytical methods

Classical methods convert the equation into simpler, solvable pieces, giving a closed form when possible and a series or an integral otherwise.

linear first order equations

Separation of variables: Reorganise by variable and reintegrate:

\frac{dy}{dx}=g(x)\,h(y)\qquad\Longrightarrow\qquad\int\frac{dy}{h(y)}=\int g(x)\,dx

For the equation dy/dx = −x/y, this gives the family of circles:

\frac{dy}{dx}=-\frac{x}{y}\qquad\Longrightarrow\qquad x^{2}+y^{2}=C

Integrating factors.

\frac{dy}{dx}+p(x)\,y=q(x),\text{multiply both sides by } \mu(x)=e^{\int p(x)\,dx}\;\Longrightarrow\;\frac{d}{dx}\bigl(\mu(x)\,y\bigr)=\mu(x)\,q(x)

For instance:

\frac{dy}{dx}+y=e^{-x}\qquad\Longrightarrow\qquad y=(x+C)e^{-x}

Characteristic equations. For a linear equation with constant coefficients, the trial solution y = e^rx reduces the differential equation to an algebraic characteristic equation. In general, for the second-order equation y″ + ay′ + by = 0:

y''+a\,y'+b\,y=0,\qquad y=e^{rx}\ \Rightarrow\ r^{2}+a\,r+b=0

For the example

y=e^{rx}\ \Rightarrow\ (r^{2}-3r+2)\,e^{rx}=0\ \Rightarrow\ r^{2}-3r+2=(r-1)(r-2)=0

whose characteristic roots r = 1 and r = 2 are real and distinct, each root contributes one exponential, and the general solution is the sum

y''-3y'+2y=0\qquad\Longrightarrow\qquad y=C_{1}e^{x}+C_{2}e^{2x}

When the roots form a complex-conjugate pair r = α ± iβ, Euler's formula merges the two complex exponentials into real oscillations, and the general solution is

r=\alpha\pm i\beta\ \Rightarrow\ y=e^{\alpha x}\bigl(C_{1}\cos\beta x+C_{2}\sin\beta x\bigr)

For the harmonic oscillator y″ + y = 0, the characteristic roots are r = ±i, so the general solution is sinusoidal:

y''+y=0\ \Rightarrow\ r^{2}+1=0\ \Rightarrow\ r=\pm i\ \Rightarrow\ y=C_{1}\cos x+C_{2}\sin x

A forced (or non-homogeneous) linear equation is one whose right-hand side is a non-zero function g(x), the forcing term, which drives the system from outside, as a driving force drives an oscillator. Its general solution is the sum of the complementary solution y_c, the general solution of the homogeneous equation that carries the two arbitrary constants, and one particular solution y_p of the full equation:

y''+p(x)\,y'+q(x)\,y=g(x),\qquad y=y_{c}+y_{p},\qquad y_{c}''+p(x)\,y_{c}'+q(x)\,y_{c}=0

Variation of parameters is a general way to construct y_p from the homogeneous solutions y₁ and y₂: replace the constants in y_c = C₁y₁ + C₂y₂ by unknown functions u₁(x), u₂(x), and impose the two conditions that make y_p = u₁y₁ + u₂y₂ solve the forced equation:

y_{p}=u_{1}(x)\,y_{1}+u_{2}(x)\,y_{2},\qquad u_{1}'y_{1}+u_{2}'y_{2}=0,\qquad u_{1}'y_{1}'+u_{2}'y_{2}'=g(x)

Solving this 2×2 linear system for u₁′ and u₂′, by dividing by the Wronskian W = y₁y₂′ − y₂y₁′, and integrating, gives the functions u₁ and u₂. For the forced example y″ − 3y′ + 2y = e^x, whose homogeneous solutions are y₁ = e^x and y₂ = e^2x:

y_{1}=e^{x},\quad y_{2}=e^{2x},\quad W=y_{1}y_{2}'-y_{2}y_{1}'=e^{3x};\qquad u_{1}'=-\frac{y_{2}\,g}{W}=-1,\quad u_{2}'=\frac{y_{1}\,g}{W}=e^{-x}

Thus u₁′ = −1 and u₂′ = e^−x, so u₁ = −x and u₂ = −e^−x. The resulting function −xe^x − e^x contains the homogeneous solution e^x, which is absorbed into C₁, leaving the particular solution y_p = −xe^x and the general solution y = C₁e^x + C₂e^2x − xe^x.

Power series. When the coefficients vary with x, a solution is sought as a power series y = Σaₙxⁿ. Substituting the series and its derivatives into the equation and equating the coefficients of like powers of x converts the differential equation into recurrence relations that fix the aₙ one after another. For the Airy equation y″ − xy = 0:

so that a₂ = 0 and, for n ≥ 1, each coefficient a_n+2 is expressed in terms of a_n−1. The two free constants a₀ and a₁ therefore generate two independent solutions:

These are, up to normalisation, the two Airy functions:

y''-xy=0\qquad\Longrightarrow\qquad y=C_{1}\operatorname{Ai}(x)+C_{2}\operatorname{Bi}(x)

Because these series converge for every x, the Airy equation, which has no elementary closed-form solution, is solved exactly in open form. About a regular singular point, where the equation's own coefficients have poles, the Frobenius method extends the same idea to series with a shifted leading power x^ρ.

Laplace transforms. The transform turns differentiation into multiplication by the transform variable s, so an initial value problem becomes an algebraic equation for Y(s):

\mathcal{L}\{y'\}=sY(s)-y(0),\qquad \mathcal{L}\{y''\}=s^{2}Y(s)-s\,y(0)-y'(0)

The procedure is mechanical: transform each term of the equation using the initial values, solve the resulting algebraic equation for Y(s), expand the answer in partial fractions, and invert each term with a table of standard transforms. For y″ + y = 1 with y(0) = y′(0) = 0, transforming both sides gives

s^{2}Y(s)+Y(s)=\mathcal{L}\{1\}=\frac{1}{s}\quad\Longrightarrow\quad (s^{2}+1)\,Y(s)=\frac{1}{s}

Solving for Y(s) and expanding in partial fractions,

Y(s)=\frac{1}{s(s^{2}+1)}=\frac{1}{s}-\frac{s}{s^{2}+1}

and inverting term by term, using the standard pairs 1/s ↔ 1 and s/(s² + 1) ↔ cos t, recovers the solution:

y''+y=1,\qquad y(0)=0,\quad y'(0)=0\qquad\Longrightarrow\qquad y(t)=1-\cos t

Separation of variables for PDEs. With the product ansatz u(x, t) = X(x)T(t), a linear PDE splits into ordinary equations tied together by a separation constant λ:

u=X(x)\,T(t)\ \Rightarrow\ \frac{X''}{X}=\frac{T'}{\alpha\,T}=-\lambda

Superposition of the product solutions then fits the initial data. For the heat equation on a rod of length L whose ends are held at temperature zero, this gives the Fourier sine series

u(x,t)=\sum_{n=1}^{\infty}b_{n}\sin\Bigl(\frac{n\pi x}{L}\Bigr)\,e^{-\alpha (n\pi/L)^{2}t}

whose terms decay at rates set by the thermal diffusivity α.

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. Euler's method replaces the exact increment with the first-order Taylor approximation:

y(x+h)=y(x)+h\,f\bigl(x,y(x)\bigr)+\mathcal{O}\bigl(h^{2}\bigr)

and 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})

Euler's method approximates a solution curve by following the local slope over short steps; smaller steps follow the exact curve more closely. Credit: Oleg Alexandrov (public domain).

The error committed in one step is proportional to h², so over the 1/h steps spanning a fixed interval the accumulated error is proportional to h. For the model problem dy/dx = y with y(0) = 1, whose exact solution is y = e^x, each Euler step multiplies the estimate by 1 + h:

y_{n+1}=(1+h)\,y_{n}\;\Longrightarrow\; y_{n}=(1+h)^{n},\qquad \lim_{h\to 0}(1+h)^{1/h}=e

so at x = 1 the estimate (1 + h)^1/h tends to e as h → 0, and the polygonal approximation converges to the exact solution. More accurate and more stable methods, such as the Runge–Kutta family, remove the low-order error terms by combining several slope evaluations per step, 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 begins with the equilibria, the points at which the system does not change, and determines their stability: a small perturbation away from a stable equilibrium decays back towards it, while the system leaves an unstable one.

The undamped pendulum

\frac{d^{2}\theta}{dt^{2}}+\frac{g}{L}\sin\theta=0

has its downward position θ = 0 and its upright position θ = π as equilibria. Written as a first-order system in the phase plane (θ, v),

\frac{d\theta}{dt}=v,\qquad \frac{dv}{dt}=-\frac{g}{L}\sin\theta,\qquad \text{equilibria at }(\theta,v)=(0,0)\ \text{and}\ (\pi,0)

their stability follows by linearising about each equilibrium:

\theta\approx 0:\ \ \frac{d^{2}\theta}{dt^{2}}+\frac{g}{L}\theta=0;\qquad \phi=\theta-\pi\approx 0:\ \ \frac{d^{2}\phi}{dt^{2}}-\frac{g}{L}\phi=0

At the downward position the linearised equation is the harmonic oscillator, and the phase plane is filled with the closed loops of periodic swinging; at the upright position the linearised equation has exponential solutions, a saddle through which the flow escapes.

As a parameter changes, equilibria can appear, merge or vanish in a bifurcation. A harvested fish population obeys the harvested logistic equation

\frac{dN}{dt}=rN\Bigl(1-\frac{N}{K}\Bigr)-H,\qquad H_{c}=\frac{rK}{4}

which keeps two equilibria while the harvest H stays below H_c = rK/4; at H_c they merge into one, and beyond it no equilibrium survives and the population collapses.

In three or more dimensions a system can settle onto a chaotic attractor: its solutions stay bounded and deterministic, yet are aperiodic and so sensitive to initial conditions that nearby states diverge rapidly. Edward Lorenz found the first example in 1963 in the system that now bears his name,

\frac{dx}{dt}=\sigma\,(y-x),\qquad \frac{dy}{dt}=x\,(\rho-z)-y,\qquad \frac{dz}{dt}=x\,y-\beta\,z

which, for σ = 10, ρ = 28 and β = 8/3, traces the famous butterfly-shaped attractor. Chaos has since been found throughout physics, biology, chemistry and economics; it is why long-term weather prediction is impossible in practice.

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. Henri Poincaré developed qualitative, geometric methods to study differential equations at the end of the 19th century to tackle the three-body problem of celestial mechanics. In 1892, Aleksandr Lyapunov gave a general definition of stability.

The growing power of numerical computing in the 20th century enabled the development of the theory of dynamical systems that now pervades the sciences. 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. Although the 3-equation system was deterministic, its solutions were aperiodic and so sensitive to initial conditions that nearby states rapidly diverged. He popularised the phenomenon as the butterfly effect.

Further reading

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