Differential equation

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Differential equation

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'''A differential equation''' is an equation in which the unknown is a function, and in which the derivatives (rates of change) of that function also appear. An ordinary equation such as $x^2 = 9$ has numbers as solutions, whereas a differential equation such as $2y+y'=0$ has functions $y(x)$ as solutions.

Most laws of nature are stated as rules about how quantities change, so differential equations appear throughout science and engineering: the swinging of a pendulum, the cooling of a hot drink, the growth of a population, and the discharge of a capacitor are all described by differential equations. This article explains what a differential equation is and how equations are classified, then works through the most common exact methods of solution, each illustrated by a concrete numerical example that needs no background knowledge beyond school algebra. It treats '''ordinary differential equations''' (ODEs), in which the unknown function depends on a single independent variable, in the most detail, and gives shorter accounts of '''partial differential equations''' (PDEs) and of the numerical, series, and qualitative methods used when no exact formula exists.

== A first example: slopes and a family of solutions ==

Take the simplest possible differential equation. Suppose the unknown is a function $y(x)$, and all we are told about it is how it changes:

$$\frac{dy}{dx}=2x$$

Here $dy/dx$ is the slope of the graph of $y$. The equation says that whatever the solution is, its slope at the point $x$ must equal $2x$.

To solve the equation is to find every function whose slope behaves this way. Integrating both sides with respect to $x$ undoes the differentiation on the left, so

$$\int\frac{dy}{dx}\,dx=\int 2x\,dx\qquad\Longrightarrow\qquad y(x)=x^{2}+C$$

where $C$ is an arbitrary constant, because differentiating $x^2 + C$ gives $2x$ for every value of $C$:

$$\frac{d}{dx}\left(x^{2}+C\right)=2x$$

The solutions are therefore not one function but a whole family of parabolas $y = x^2 + C$, one for each choice of $C$, each a vertical shift of the others. This family is called the '''general solution'''.

How can we arrive at a '''particular solution''', that is, one specific function of the family? An extra piece of information is required.

Suppose we know that when $x = 0$, $y = 3$. Substituting those numbers in we have:

$$3=0^{2}+C\qquad\Longrightarrow\qquad C=3$$

Therefore

$$y = x^{2} + 3$$

Such an extra condition is called an '''initial condition'''.

== Classifying differential equations ==

A few labels do most of the work in deciding how to attack an equation. Three questions matter: how many derivatives appear (the ''order''), whether the unknown function enters linearly (''linearity''), and how many independent variables are involved (''ordinary or partial''). Equations that fit no convenient label are usually handled numerically or qualitatively, as described in the final section of this article.

=== Order ===

The '''order''' of a differential equation is the order of the highest derivative that appears in it. The equation $dy/dx = 2x$ of the previous section is first order.

Newton's second law of motion is the standard second-order example: the acceleration of a body, the second derivative of its position, is proportional to the force acting on it:

{{#content:Q1583}}

where $x(t)$ is the position of a body of mass $m$ and $F$ is the net force.

In general, the general solution of an nth-order equation contains $n$ arbitrary constants, so $n$ extra conditions are needed to single out one particular solution. When the equation has the special form $y^{(n)}=f(x)$, the constants appear naturally, one at each of the $n$ integrations needed to undo the derivatives; the free-fall example below shows the pattern for $n = 2$.

Take a free-falling object, for example. Ignoring air resistance, the only force is gravity

$$F_g = -mg$$

By Newton's second law

$$m\frac{d^{2}x}{dt^{2}}=-mg\qquad\Longrightarrow\qquad\frac{d^{2}x}{dt^{2}}=-g$$

with $g \approx 9.8\ \mathrm{m\,s^{-2}}$ the acceleration of free fall. Integrating both sides once gives the velocity, and introduces the constant $v_{0}$, the speed at time $t = 0$:

$$\frac{dx}{dt}=-gt+v_{0}$$

Integrating again gives the height, and introduces a second constant, $x_{0}$, the height at $t = 0$:

$$x(t)=-\frac{g}{2}\,t^{2}+v_{0}t+x_{0}$$

Two conditions, the initial height and the initial velocity, are needed to fix both constants.

A concrete check: a ball dropped from rest ($v_{0} = 0$) at a height of $19.6\ \mathrm{m}$ hits the ground when $x(t) = 0$:

$$0=19.6-4.9\,t^{2}\qquad\Longrightarrow\qquad t=\sqrt{19.6/4.9}=2\ \text{seconds}$$

Notice how a differential equation can predict the future: the two initial conditions fix the whole trajectory.

=== Linearity and homogeneity ===

A differential equation is '''linear''' when the unknown function and its derivatives appear only to the first power and are never multiplied together (multiplying by functions of the independent variable is allowed). A linear first-order equation can always be written as

$$\frac{dy}{dx}+p(x)\,y=q(x)$$

Furthermore, a linear equation is called '''homogeneous''' when $q(x)=0$; it then takes the form

$$\frac{dy}{dx}+p(x)\,y=0$$

The equations

$$\frac{dy}{dx}=y^{2},\qquad \frac{d^{2}\theta}{dt^{2}}+\sin\theta=0$$

are therefore nonlinear: the first contains the square of the unknown function, the second the sine of it.

Linearity and homogeneity matter because homogeneous linear equations obey the '''superposition principle'''. If $y_{1}$ and $y_{2}$ both solve a homogeneous linear equation, then any combination $c_{1}y_{1} + c_{2}y_{2}$ of those functions solves it too. This is why solutions of linear equations can be added together, a property used repeatedly later in this article (for example, to build the solution of the heat equation from simple building blocks).

Nonlinear equations do not obey the superposition principle. If $y_{1}$ and $y_{2}$ solve $dy/dx = y^{2}$, their sum does not.

In general, linear equations can be solved systematically, whereas most nonlinear equations cannot; the last section of this article describes what is done with those.

=== Ordinary and partial ===

An '''ordinary differential equation''' involves a function of a single independent variable, as in all the examples so far. A '''partial differential equation''' (PDE) involves a function of several independent variables, together with its partial derivatives. For example, the temperature $u(x, t)$ of a metal bar, which depends on the position $x$ along the bar and on the time $t$, satisfies the '''heat equation'''

{{#content:Q1590}}

where the constant $\alpha$ measures how quickly heat spreads: a spot that is much warmer than its neighbours (large second derivative $\partial^2 u/\partial x^2$) warms or cools quickly.

== Slope fields: visualising solutions ==

A first-order equation solved for its derivative has the general form

{{#content:Q1581}}

It assigns to every point $(x, y)$ of the plane the slope that any solution curve passing through that point must have there. Drawing a short line segment with exactly that slope at many points produces a '''direction field''' (or slope field) for the equation.

[[File:Slope field of exponential growth.png|thumb|Direction field of $dy/dx = y$. Each short segment shows the slope that a solution must have there, and the drawn curves follow the field. Credit: jjbeard (public domain).]]

A solution curve must be tangent to the field everywhere it passes, like a boat pushed by a current whose direction depends on where the boat is. The field therefore displays the whole solution family at a glance.<ref>{{#cite:Q1576}}</ref> Numerical methods, such as [[Euler's method]], work by following such arrows step by step: from a starting point, read the slope of the arrow there, take a small step in that direction, read the new arrow, and repeat.

== Solving differential equations ==

There is no single recipe that solves every differential equation. The practical approach is to recognise which family an equation belongs to, then apply that family's method. The subsections below work through the families that appear most often, each stated in general and then illustrated with numbers; the last subsection collects the routes taken when none of these formulas applies.

=== Separation of variables ===

A first-order equation is '''separable''' when the right-hand side splits into a factor that depends only on $x$ and a factor that depends only on $y$:

{{#content:Q1612}}

The arrow shows the whole method: divide both sides by $h(y)$ and integrate, so that all the $y$'s are on one side and all the $x$'s on the other. The two worked examples below carry this out completely, and both lead to the same conclusion: quantities whose rate of change is proportional to their own size change exponentially.

==== Worked example: exponential growth and decay ====

Many quantities change at a rate proportional to their own size. A population with unlimited food grows faster the larger it is, money in a bank earns interest in proportion to the balance, and the number of radioactive atoms left decreases in proportion to how many remain. Writing the constant of proportionality as $k$,

{{#content:Q1584}}

where $k$ is a constant. To solve it, divide both sides by $y$ and integrate both sides with respect to $t$:

$$\frac{1}{y}\frac{dy}{dt}=k\qquad\Longrightarrow\qquad\int\frac{dy}{y}=\int k\,dt$$

The left-hand integral is $\ln|y|$, so

$$\ln|y| = kt + C$$

Exponentiating both sides (raising $e$ to the power of each side) removes the logarithm:

$$|y|=e^{kt+C}=e^{C}e^{kt}$$

The sign of $y$ never changes, so the absolute value can be dropped by absorbing the sign into the constant. Writing $y_{0}$ for the value at $t = 0$, we obtain

{{#content:Q1585}}

With numbers: suppose €1000 is deposited in an account paying 5% per year with interest added continuously, so $k = 0.05$ per year and $y_{0} = 1000$. The balance is $y(t) = 1000\,e^{0.05t}$, and after 10 years

$$y(10)=1000\,e^{0.05\times 10}=1000\,e^{0.5}\approx 1648.7$$

so the deposit has grown to about €1649. To find when it doubles, set $y(t) = 2000$ and solve:

$$1000\,e^{0.05t}=2000\qquad\Longrightarrow\qquad e^{0.05t}=2\qquad\Longrightarrow\qquad t=\frac{\ln 2}{0.05}\approx 13.9\ \text{years}$$

For $k < 0$ the same formula describes decay rather than growth. Radioactive substances are usually described by their '''half-life''', the time in which half the atoms decay; setting $y(t)=y_{0}/2$ gives $t_{1/2}=(\ln 2)/(-k)$.

==== Worked example: Newton's law of cooling ====

A hot object left in a cooler room cools at a rate proportional to how much hotter it is than the room. This is Newton's law of cooling,

{{#content:Q1586}}

where $T(t)$ is the temperature of the object, $T_{a}$ is the constant room temperature, and the positive constant $k$ measures how easily heat escapes.

Separating variables means bringing everything that involves $T$ to the left:

$$\frac{1}{T-T_{a}}\frac{dT}{dt}=-k\qquad\Longrightarrow\qquad\int\frac{dT}{T-T_{a}}=\int -k\,dt$$

Integrating both sides gives

$$\ln|T-T_{a}|=-kt+C$$

Exponentiating both sides,

$$|T-T_{a}|=e^{-kt+C}=e^{C}e^{-kt}$$

The difference $T - T_{a}$ keeps its sign: positive while the object cools, negative while it warms towards a warmer room. Absorbing the sign into the constant and fixing it with the initial temperature $T(0)=T_{0}$, we obtain

$$T(t)=T_{a}+\left(T_{0}-T_{a}\right)e^{-kt}$$

So the temperature does not decay exponentially, but the temperature gap $T - T_{a}$ does.

With numbers: suppose a drink at $80\,^{\circ}\mathrm{C}$ is left in a room at $20\,^{\circ}\mathrm{C}$, and measurement shows $k = 0.1$ per minute. Then

$$T(t)=20+\left(80-20\right)e^{-0.1t}=20+60\,e^{-0.1t}$$

The gap starts at $60\,^{\circ}\mathrm{C}$ and shrinks by the factor $e^{-0.1t}$ each minute; it halves every $(\ln 2)/0.1 \approx 6.9$ minutes. To find when the drink reaches $40\,^{\circ}\mathrm{C}$, put $T(t)=40$:

$$40=20+60\,e^{-0.1t}\qquad\Longrightarrow\qquad e^{-0.1t}=\frac{1}{3}\qquad\Longrightarrow\qquad t=10\ln 3\approx 11\ \text{minutes}$$

In about 11 minutes the drink is two-thirds of the way from $80\,^{\circ}\mathrm{C}$ down to the room's $20\,^{\circ}\mathrm{C}$, and it approaches the room temperature without ever quite reaching it.<ref>{{#cite:Q1576}}</ref>

=== First-order linear equations: the integrating factor ===

The two equations just solved are separable as well as linear. Many first-order equations, however, are linear but not separable, and for those there is a general method based on the '''integrating factor'''.

Recall from the classification section that a linear first-order equation has the form

$$\frac{dy}{dx}+p(x)\,y=q(x)$$

The idea is to multiply both sides by a function $\mu(x)$, chosen so that the left-hand side becomes the derivative of a single product $\mu(x)\,y$, which can then be integrated in one stroke. Expanding the product rule,

$$\frac{d}{dx}\bigl(\mu(x)\,y\bigr)=\mu\,\frac{dy}{dx}+\frac{d\mu}{dx}\,y$$

Multiplying the equation by $\mu$ would give $\mu\,\frac{dy}{dx}+\mu p\,y$ on the left. For the two to agree, the coefficient of $y$ must match, so $\mu$ must satisfy $\mu' = p\,\mu$. This is itself a separable equation, solved exactly as in the previous subsection:

$$\frac{d\mu}{dx}=p(x)\,\mu\qquad\Longrightarrow\qquad\frac{d\mu}{\mu}=p(x)\,dx\qquad\Longrightarrow\qquad\ln\mu=\int p(x)\,dx\qquad\Longrightarrow\qquad\mu(x)=e^{\int p(x)\,dx}$$

(any constant of integration may be taken as zero, since multiplying $\mu$ by a constant changes nothing). Multiplying the original equation by this $\mu(x)$,

{{#content:Q1613}}

and because the left-hand side is now a single derivative, both sides integrate immediately:

$$\mu(x)\,y=\int \mu(x)\,q(x)\,dx + C$$

Worked example. Solve $y' + y = e^{-x}$. Here $p = 1$, so the integrating factor is

$$\mu=e^{\int 1\,dx}=e^{x}$$

Multiplying both sides by $e^{x}$,

$$e^{x}y'+e^{x}y=e^{x}e^{-x}=1$$

The left-hand side is $(e^{x}y)'$, so

$$\left(e^{x}y\right)'=1\qquad\Longrightarrow\qquad e^{x}y=x+C\qquad\Longrightarrow\qquad y=(x+C)e^{-x}$$

{{#content:Q1607}}

An initial condition fixes the constant: if $y(0)=2$, then $2=(0+C)e^{0}$, so $C=2$ and $y=(x+2)e^{-x}$.<ref>{{#cite:Q1576}}</ref><ref>{{#cite:Q1577}}</ref>

=== Constant-coefficient linear equations of order two ===

The methods so far solve first-order equations. The most important second-order family is the linear equation with constant coefficients,

$$y''+a\,y'+b\,y=0$$

which models systems pulled back towards a resting state: a mass on a spring, a pendulum swinging through small angles, and an electric circuit containing a capacitor and an inductor. The key idea is to try an exponential solution $y=e^{rx}$, because the derivative of an exponential is again a multiple of itself, so substituting turns the differential equation into an ordinary algebraic equation:

{{#content:Q1644}}

To see where the arrow comes from, substitute $y=e^{rx}$ together with $y'=re^{rx}$ and $y''=r^{2}e^{rx}$ into the equation:

$$r^{2}e^{rx}+a\,r\,e^{rx}+b\,e^{rx}=\left(r^{2}+a\,r+b\right)e^{rx}=0$$

Since $e^{rx}$ is never zero, divide both sides by it:

$$r^{2}+a\,r+b=0$$

This is the '''characteristic equation'''. It is an ordinary quadratic equation, and the form of the solution depends on the kind of roots it has:

* '''Two distinct real roots''', $r_{1} \neq r_{2}$: the general solution is $y=C_{1}e^{r_{1}x}+C_{2}e^{r_{2}x}$, a sum of two exponentials.
* '''One repeated real root''', $r$: the general solution is $y=\left(C_{1}+C_{2}x\right)e^{rx}$; the extra factor $x$ supplies the second arbitrary constant.
* '''A complex conjugate pair''', $r=\alpha \pm i\beta$: the general solution is $y=e^{\alpha x}\left(C_{1}\cos\beta x + C_{2}\sin\beta x\right)$, which oscillates, growing or fading in size according to the sign of $\alpha$.

A quick numerical example of the first case: solve $y''-3y'+2y=0$. Substituting $y=e^{rx}$ gives $r^{2}-3r+2=0=(r-1)(r-2)$, so the roots are $1$ and $2$, and

{{#content:Q1608}}

Each term works, as can be checked directly: for $y=e^{x}$, one has $y''-3y'+2y=(1-3+2)e^{x}=0$, and similarly for $e^{2x}$.

==== Example: the harmonic oscillator (a mass on a spring) ====

Some quantities do not settle towards a level but swing back and forth. The complex-root case above is not an exotic exception: it is exactly what happens for the most common oscillating system of all. Consider a mass $m$ attached to a spring. If the spring is displaced a distance $x$ from its rest position, it pulls back with a force $-kx$ proportional to the displacement (Hooke's law), where the spring constant $k$ measures how stiff the spring is. Newton's second law therefore gives

$$m\frac{d^{2}x}{dt^{2}}=-kx\qquad\Longrightarrow\qquad\frac{d^{2}x}{dt^{2}}+\frac{k}{m}x=0$$

Writing $\omega_{0}^{2} = k/m$, this becomes the '''harmonic oscillator equation''':

{{#content:Q1588}}

Its characteristic equation is $r^{2}+\omega_{0}^{2}=0$, with the purely imaginary roots $r=\pm i\omega_{0}$: this is the complex-conjugate case above with $\alpha=0$ and $\beta=\omega_{0}$. The equation asks for a function whose second derivative is a negative constant multiple of itself, and the sine and cosine have exactly this property. Differentiating $\cos(\omega_{0}t)$ twice brings out two factors of $\omega_{0}$ and a minus sign:

$$\frac{d}{dt}\cos(\omega_{0}t)=-\omega_{0}\sin(\omega_{0}t),\qquad \frac{d^{2}}{dt^{2}}\cos(\omega_{0}t)=-\omega_{0}^{2}\cos(\omega_{0}t)$$

so $x = \cos(\omega_{0}t)$ solves the equation, and so does $x = \sin(\omega_{0}t)$. Because the equation is linear and homogeneous, the superposition principle applies and the general solution is

$$x(t)=A\cos(\omega_{0}t)+B\sin(\omega_{0}t)$$

The two constants $A$ and $B$ are fixed by the initial displacement and the initial velocity, exactly as the order of the equation requires.

[[File:Simple harmonic motion animation.gif|thumb|A mass on a spring executes simple harmonic motion: the solution of the harmonic oscillator equation is a sinusoid of fixed amplitude and frequency. Credit: Evil saltine (public domain).]]

A concrete example. Take a mass of $2\ \mathrm{kg}$ on a spring with $k = 8\ \mathrm{N/m}$, so that $\omega_{0} = \sqrt{k/m} = \sqrt{4} = 2$ radians per second. Pull the mass $10\ \mathrm{cm}$ out and release it from rest: the initial velocity is zero, so $B = 0$, and $x(t) = 0.10\cos(2t)$ metres. The motion repeats after one period

$$P=\frac{2\pi}{\omega_{0}}=\pi\ \text{seconds}\approx 3.14\ \text{s}$$

so the mass returns to its starting point roughly every 3.14 seconds. One second after release, measuring angles in radians,

$$x(1)=0.10\cos(2)\approx 0.10\times(-0.416)\approx -0.042\ \text{m}$$

about 4 cm on the other side of the rest position. This kind of motion, a sinusoid of fixed amplitude, is called '''simple harmonic motion''', and the oscillator equation governs not only springs but pendulums (for small swings), electric circuits, and the vibrations of molecules.<ref>{{#cite:Q1577}}</ref>

=== Partial differential equations: separating variables in the heat equation ===

Exact formulas for partial differential equations exist only for a few simple, usually linear, problems, but these are precisely the problems at the heart of physics and engineering. The standard technique is again separation of variables, but now the variables to be separated are the independent variables themselves, here the position $x$ and the time $t$.

Return to the metal bar of the classification section, this time of length $L$ with its sides insulated, so that heat flows only along the bar, and with both ends held at temperature $0$. Its temperature $u(x,t)$ satisfies the heat equation, reproduced here:

$$\frac{\partial u}{\partial t}=\alpha\,\frac{\partial^{2}u}{\partial x^{2}}$$

Separation of variables looks for a solution that splits into a product of a function of $x$ alone and a function of $t$ alone:

$$u(x,t)=X(x)\,T(t)$$

Substituting into the heat equation gives $X\,T'=\alpha\,X''\,T$. Divide both sides by $\alpha X T$:

{{#content:Q1622}}

The left-hand side now depends only on $t$ and the right-hand side only on $x$. Since $x$ and $t$ can vary independently, the only way two functions of different variables can be equal for all $x$ and $t$ is that both sides equal the same constant, written above as $-\lambda$. This splits the partial differential equation into two ordinary ones:

* $T'=-\alpha\lambda\,T$: exponential decay, the growth equation of this article with $k=-\alpha\lambda<0$, giving $T=e^{-\alpha\lambda t}$;
* $X''=-\lambda X$: the function whose second derivative is a negative constant multiple of itself, giving sines and cosines of $\sqrt{\lambda}\,x$.

The ends of the bar are held at temperature $0$, so $u(0,t)=u(L,t)=0$, which forces $X(0)=X(L)=0$. The condition $X(0)=0$ removes the cosine, and $X(L)=0$ forces $\sin(\sqrt{\lambda}\,L)=0$, so $\sqrt{\lambda}\,L=n\pi$ for a positive whole number $n$. Thus only the special values $\lambda=(n\pi/L)^{2}$ are allowed, each giving one mode

$$u_{n}(x,t)=\sin\left(\frac{n\pi x}{L}\right)e^{-\alpha(n\pi/L)^{2}t}$$

a half-wave, or several half-waves, of a sine curve that sinks towards zero as heat leaks out of the ends. Because the heat equation is linear and homogeneous, superposition applies: any sum of modes is again a solution, and the general solution is

{{#content:Q1611}}

The numbers $b_{n}$ are fixed by the initial temperature profile $u(x,0)$ at $t=0$; writing an arbitrary initial profile as such a sum of sine waves is exactly what Fourier's sine series does.

Two features are worth noticing. First, every mode decays exponentially, so the bar does not oscillate: heat spreads and the temperature evens out. Second, modes with more wiggles decay faster, because their rate $-\alpha(n\pi/L)^{2}$ is larger in magnitude, so after a short time only the $n=1$ mode, one smooth half-sine across the whole bar, is left.

A concrete check with numbers: take a one-metre iron bar (iron's thermal diffusivity is $\alpha \approx 2.3\times10^{-5}\ \mathrm{m^{2}\,s^{-1}}$) whose initial temperature is $u(x,0)=100\sin(\pi x/L)$, that is, $0\,^{\circ}\mathrm{C}$ at the ends and $100\,^{\circ}\mathrm{C}$ in the middle. Only the $n=1$ mode is present, so

$$u(x,t)=100\,\sin\left(\frac{\pi x}{L}\right)e^{-\alpha\pi^{2}t/L^{2}}$$

With $L=1\ \mathrm{m}$, the decay rate is $\alpha\pi^{2}\approx 2.3\times10^{-5}\times9.87\approx 2.3\times10^{-4}$ per second. At the centre of the bar, where the sine equals $1$,

$$u\!\left(\tfrac{1}{2},t\right)=100\,e^{-2.3\times10^{-4}\,t}$$

After one hour ($t=3600\ \mathrm{s}$) the exponent is $2.3\times10^{-4}\times3600\approx0.82$, so the centre has cooled to $100\,e^{-0.82}\approx44\,^{\circ}\mathrm{C}$, and it reaches $50\,^{\circ}\mathrm{C}$ after $t=(\ln 2)/(2.3\times10^{-4})\approx 3050\ \mathrm{s}$, about 51 minutes. The heat has flowed out through the ends, exactly the phenomenon Fourier set out to describe.<ref>{{#cite:Q1579}}</ref>

=== When no formula is possible: numerical, series, and qualitative methods ===

The families above, separable and linear with convenient coefficients, cover a great many practical equations, but they are a small minority of all differential equations. There is no guarantee that a given equation, especially a nonlinear one, has a solution expressible in terms of the familiar functions at all. Such equations are studied in one of three ways:

* '''Numerical approximation''', when numbers are what is needed: [[Euler's method]] steps forward along the slope field, replacing the exact solution by a polygonal path, and its refinements underlie most computer simulations;
* '''[[Qualitative methods]]''', when behaviour matters more than numbers: the solutions are studied through the slope field, equilibria, stability, and long-term behaviour, without ever writing a formula;
* '''Series and integral transforms''', for special but important cases: when an equation is linear, a solution can sometimes be written as an infinite series ([[Power series]]) or recovered from an algebraic equation by the [[Laplace transform]].

Choosing between these routes is part of the art of applying mathematics, and the diagram below summarises the decision process of this whole article.<ref>{{#cite:Q1576}}</ref><ref>{{#cite:Q1578}}</ref>

<uml type="uml">
@startuml
skinparam monochrome true
start
:You have a differential equation;
if (Is it first order and separable?\ny' = g(x) h(y)?) then (yes)
:Separate and integrate:\n∫ dy/h(y) = ∫ g(x) dx;
else (no)
if (Is it first order and linear?\ny' + p(x) y = q(x)?) then (yes)
:Multiply by the integrating factor\nμ = e^{∫ p dx}, then integrate;
else (no)
if (Is it second order, linear, with constant coefficients?\ny'' + a y' + b y = 0?) then (yes)
:Solve the characteristic equation\nr² + a r + b = 0;
else (no)
if (Is it a linear PDE on a simple\nshape, e.g. the heat equation?) then (yes)
:Separate the variables:\nu(x,t) = X(x) T(t);
else (no)
if (Are approximate numbers enough?) then (yes)
:Step forward numerically\n(article: Euler's method);
else (no)
if (Is the equation linear?) then (yes)
:Power series or Laplace transform\n(articles: Power series, Laplace transform);
else (no)
:Study behaviour without formulas:\nequilibria, stability, chaos\n(article: Qualitative methods);
endif
endif
endif
endif
endif
endif
stop
@enduml
</uml>

As the diagram shows, the exact methods occupy only the first few branches; most equations encountered in research are handled by the routes at the bottom of the diagram, each of which has its own article.

== A short history ==

Differential equations are as old as the calculus itself, because the calculus is the mathematics of change: the laws of physics say how quantities change, and predicting the future means undoing those changes. When Newton published his laws of motion and of gravitation in the ''Principia'' in 1687, the equations he needed were differential equations, and he solved them by geometry and by infinite series. Newton had his own notation for rates of change, but it was Leibniz's $dy/dx$, first written in the 1670s, that survived: it displays the whole equation on the page, and it is the notation used throughout this article.<ref>{{#cite:Q1577}}</ref>

[[File:Isaac Newton portrait.jpg|thumb|left|Isaac Newton (portrait after Godfrey Kneller, 1689). The laws of motion and of gravitation published in the ''Principia'' (1687) are differential equations. Credit: James Thronill after Godfrey Kneller (public domain).]]

The scattered tricks of the early calculus became a subject when [[Person:Leonhard Euler|Leonhard Euler]] took them up in the middle decades of the 18th century. He recognised that a linear equation with constant coefficients is solved by substituting $y=e^{rx}$, turning calculus into algebra; he developed series solutions; and, for equations that resisted formulas, he invented the step-by-step numerical scheme, described in this article, that still bears his name. Most of the exact methods above descend from his work.<ref>{{#cite:Q1577}}</ref>

[[File:Leonhard Euler portrait.jpg|thumb|Leonhard Euler (portrait by Jakob Emanuel Handmann, 1753). Credit: Jakob Emanuel Handmann (public domain).]]

Meanwhile physics began asking for equations with more than one independent variable. A plucked string takes a shape that depends on position along the string and on time, and in 1747 [[Person:Jean le Rond d'Alembert|Jean le Rond d'Alembert]] wrote down the wave equation for it and solved it, showing that its solutions are two waves travelling in opposite directions. Heat conduction posed a subtler problem, because the initial temperature of a bar can have any shape at all. [[Person:Joseph Fourier|Joseph Fourier]] derived the heat equation from the physics of conduction and, to solve it, had to express an arbitrary initial profile as a sum of sine ripples. His ''Théorie analytique de la chaleur'' of 1822 turned separation of variables into a cornerstone of applied mathematics, and the Fourier series invented for the purpose now appears wherever signals are analysed, from acoustics to image compression.<ref>{{#cite:Q1579}}</ref>

The exact formulas, however, have their limits, and the history of the subject since the late 19th century is largely the story of what to do when no formula exists. Studying the three-body problem of celestial mechanics, [[Person:Henri Poincaré|Henri Poincaré]] realised that the shape of the motion can be understood without solving the equations, founding the qualitative theory of dynamical systems. The electronic computer then made the numerical route routine: approximate the solution step by step, as Euler's method does, refining the steps until the error is acceptable. The two strands met in 1963, when the meteorologist [[Person:Edward Lorenz|Edward Lorenz]] found that a simple system of three differential equations, meant to model atmospheric convection, behaved chaotically: the equations were deterministic, yet their solutions were aperiodic and so sensitive to initial conditions that long-term weather prediction is impossible in practice. Each of these later routes, qualitative study, numerical stepping, series, and transforms, is the subject of its own article.<ref>{{#cite:Q1578}}</ref>

== References ==

<references/>

== Further reading ==

* [https://en.wikipedia.org/wiki/Differential_equation Differential equation], Wikipedia
* [https://mathworld.wolfram.com/DifferentialEquation.html Differential Equation], Wolfram MathWorld
* [https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/ 18.03SC Differential Equations], MIT OpenCourseWare
* [https://mathshistory.st-andrews.ac.uk/Biographies/Euler/ Leonhard Euler], MacTutor History of Mathematics

Person:Joseph Fourier

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{{Person}}

== Overview ==

French mathematician and physicist; derived the heat equation and solved it with trigonometric series, founding Fourier analysis.

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Person:Jean le Rond d'Alembert

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== Overview ==

French mathematician, physicist and philosopher; wrote down and solved the wave equation of the vibrating string in the 1740s.

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File:Heat equation numerical solution.gif

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Numerical solution of the one-dimensional heat equation showing how an initial temperature profile spreads and flattens over time. Credit: Kondensat (CC BY-SA 3.0), Wikimedia Commons.

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Plot of the displacement of a damped harmonic oscillator for the three cases: underdamped, critically damped and overdamped (initial displacement 1, released from rest). Credit: Geek3 (CC BY 3.0), Wikimedia Commons.

Differential equation

2026-09-05T13:53:54Z

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Power series

2026-09-05T13:31:13Z

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'''A power series''' is an infinite sum with the general form

{{#content:Q1667}}

The numbers $a_0$, $a_1$, $a_2$, … are constants, known as the '''coefficients''', and $c$, also a constant, is known as the '''centre''' of the series.

== The simplest form: The geometric series ==

The simplest power series to study is the one in which every coefficient is 1 and the centre is 0:

$$1 + x + x^2 + x^3 + \cdots$$

As seen, each term multiplies the previous one by $x$. This is known as a '''geometric series'''.

For a geometric series, we can calculate the partial sum of the first $N + 1$ terms, $S_N = 1 + x + x^2 + \cdots + x^N$:

Multiply both sides of this equation by $(1 - x)$ and eliminate the brackets:

$$(1-x)S_{N}=(1-x)(1+x+x^{2}+\cdots+x^{N})=(1+x+x^{2}+\cdots+x^{N})-(x+x^{2}+\cdots+x^{N+1})=1-x^{N+1}$$

Dividing both sides again by $(1 - x)$, we therefore have

$$S_{N}=\frac{1-x^{N+1}}{1-x}\qquad(x\neq 1)$$ (1)

When $N \to \infty$, we have

$$\sum_{n=0}^{\infty}x^{n}=\lim_{N \to \infty}S_N=\lim_{N \to \infty}\frac{1-x^{N+1}}{1-x}\qquad(x\neq 1) $$

* '''When $-1 < x < 1$: the series converges.''' As $N$ grows, the number $x^{N+1}$ shrinks towards 0, so the partial sums approach a definite number:

$$\sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x}\qquad\text{whenever }-1<x<1$$

With $x = 0.1$, for instance, the partial sums run 1; 1.1; 1.11; 1.111; …, settling on 1.111…, and indeed $1/(1 - 0.1) = 1/0.9 = 10/9 = 1.111\ldots$

* '''When $x = 1$: the series diverges.''' Notice (1) breaks down here, because $1 - x = 0$ and we cannot divide by 0. The sum is, however, easy to calculate: every term equals 1, so $\lim_{N \to \infty} S_N = \lim_{N \to \infty} (N + 1) = \infty$

* '''Case $x = -1$: the series diverges by oscillation.''' The partial sums run 1, 0, 1, 0, 1, … and never settle on a single number. The formula $1/(1 - x)$ would give $1/2$ at $x = -1$, but notice that is the average of 1 and 0. The formula (1) does not guarantee convergence.

* '''Case $|x| > 1$: the series diverges, as expected.''' When $x > 1$, the infinite sum is the sum of an infinite number of increasingly large positive numbers, which goes to infinity. When $x < -1$, the infinite sum oscillates between positive and negative infinity, as it is dominated by the last term, which may be positive or negative.

=== Real-world application of geometric series ===

If someone promises to pay you €100 each year until the end of your life on earth, with the first payment made today, how much is that promise worth?

On first thought, this promise is worth quite a lot of money, especially if you count on living for decades more. However, you must take into account inflation, which in Europe averages out to about 5% per year. The €100 paid today is worth its full €100, a payment of €100 one year from now is worth only €$100/1.05$ in today's money, and a payment in two years is worth only €$(100)/(1.05)^2$ in today's money, and so on. A payment of €100 every year, forever, is therefore worth today

$$\sum_{n=0}^{\infty}\frac{100}{1.05^{n}}=2100$$

It is still a handsome sum, if you can live on forever.

== Radius of convergence of a power series ==

In the previous section, we have seen that even the simplest power series, a geometric series, may or may not converge depending on the value of $x$. In general, for a power series centred at $c$, there is a number $R \geq 0$, called the '''radius of convergence''', such that:

* the series converges for every $x$ with $|x - c| < R$ (closer to the centre than $R$);
* the series diverges for every $x$ with $|x - c| > R$.

The set $|x - c| < R$, an interval of length $2R$ centred at $c$, is called the '''interval of convergence of the series'''.

For the two boundary points where $|x - c| = R$, the convergence is uncertain and depends on specific characteristics of a given power series.

In general, consecutive terms of a power series $a_{n+1} (x - c)^{n+1}$ and $a_n (x - c)^n$ have magnitudes in the ratio $|a_{n+1}/a_n| \cdot |x - c|$. If, in the long run, this ratio stays below 1, intuitively the series behaves like a convergent geometric series; if the ratio stays above 1, terms grow larger and larger and the series diverges. Therefore we have

$$R=\lim_{n\to\infty}\left|\frac{a_{n}}{a_{n+1}}\right|\quad\text{when the limit exists}$$

A quick sanity check with geometric series: For the geometric series, every coefficient is 1, so the ratio is 1 and $R = 1$, matching the interval $-1 < x < 1$ found by direct calculation in the section above.

Two special cases exist:

* '''Infinite radius, $R = \infty$:''' the series converges for every $x$. The exponential and sine series in the next section are the standard examples.
* '''Zero radius, $R = 0$:''' the series converges only at the centre itself. For instance the series $1 + x + 2!\,x^2 + 3!\,x^3 + \cdots$ (whose coefficient of $x^n$ is $n!$) has $|a_n/a_{n+1}| = n!/(n + 1)! = 1/(n + 1) \to 0$, so $R = 0$; for any $x \neq 0$ the terms eventually grow without bound.

== The exponential and sine series ==

The power series can be used to approximate known functions that are difficult to calculate directly, such as $e^x$ or $\sin x$.

Imagine we would like to approximate some function $f(x)$ with a power series. We would therefore write

$$f(x)=\sum_{n=0}^{\infty}a_{n}(x-c)^{n}$$

where $c$ and $a_n$ are unknown.

Taking the $k$-th derivative of $f(x)$, we have

$$f^{(k)}(x)=\sum_{n=k}^{\infty}\frac{n!}{(n-k)!}\,a_{n}\,(x-c)^{n-k}$$

Evaluating at $x=c$, we have

$$a_{k}=\frac{f^{(k)}(c)}{k!}$$

Generalising to every $n$, we obtain

{{#content:Q1688}} (2)

=== The exponential series ===

The exponential function $e^x$ has defining properties $\frac{d e^x}{dx}=e^x$ and $e^0 = 1$. Consequently, choosing $c=0$ we have

$$f^{(n)}(0) = 1$$

for every $n$. From (2) we have $a_n = 1/n!$, so

{{#content:Q1684}}

Intuitively, the factorial $n! = 1\cdot 2\cdot 3\cdot\cdots\cdot n$ in the denominator grows much faster than any power function towards infinity, so the terms shrink and the series should converge for every $x$.

Applying the ratio test confirms $R = \infty$:

$$R=\lim_{n\to\infty}\left|\frac{a_{n}}{a_{n+1}}\right|=\lim_{n\to\infty}\frac{(n+1)!}{n!}=\lim_{n\to\infty}(n+1)=\infty$$

A quick sanity check: setting $x = 1$ in the series gives the number $e$ itself: $e = 1 + 1 + 1/2 + 1/6 + 1/24 + \cdots = 2.718\,281\,828\ldots$

=== The sine series ===

Again choosing $c=0$:

{{#content:Q1686}}

Apply once again the ratio test:

$$\lim_{k \to \infty} \frac{\frac{x^{2k+1}}{(2k+1)!}}{\frac{x^{2k+3}}{(2k+3)!}}=\lim_{k \to \infty}\frac{(2k+2)(2k+3)}{x^{2}}= \infty$$

which means the series converges as expected, to $\sin x$.

== The binomial series ==

One more family of power series is useful enough to know by name. For any fixed exponent $p$,

{{#content:Q1689}}

== Power series solutions of differential equations ==

A further application of power series is to solve differential equations whose coefficients vary with $x$. Such equations rarely have solutions built from a finite combination of familiar functions, but a solution can sometimes be written down as a power series. Substituting the series into the equation and equating the coefficients of like powers of $x$ turns the differential equation into recurrence relations that fix the coefficients one after another; the constants left free by the recurrence are exactly the arbitrary constants of the equation, and the resulting series solves the equation exactly on its interval of convergence.<ref>{{#cite:Q1577}}</ref><ref>{{#cite:Q1576}}</ref>

=== Case 1: expanding about an ordinary point, the Airy equation ===

The Airy equation

$$y^{\prime\prime} - xy = 0$$ (3)

is named after [[Person:George Biddell Airy|George Biddell Airy]], who employed it in 1838 while studying the intensity of light near a caustic. Its solutions cannot be written as finite combinations of elementary functions, so it is the standard test case for series methods.

Setting

$$y=C\sum_{n=0}^{\infty}a_{n}x^{n}$$

where $a_n$ are unknown coefficients. We have therefore

$$\quad\Rightarrow\quad y^{\prime\prime}=C\sum_{n=0}^{\infty}(n+2)(n+1)\,a_{n+2}\,x^{n},\qquad xy=C\sum_{n=1}^{\infty}a_{n-1}\,x^{n}$$

Substituting these two expressions back into (3) and collecting terms, noting that the equality must hold for every value of $x$, gives

$$\quad a_{2}=0,\qquad (n+2)(n+1)\,a_{n+2}=a_{n-1}\qquad(n\geq 1)$$

We can thereby express all the coefficients in terms of $a_0$ and $a_1$:

$$a_3 = a_0/(3\cdot 2) = a_0/6, \qquad a_6 = a_3/(6\cdot 5) = a_0/180,$$

$$a_4 = a_1/(4\cdot 3) = a_1/12, \qquad a_7 = a_4/(7\cdot 6) = a_1/504,$$

$$a_5 = a_2/(5\cdot 4) = 0, \qquad a_8 = a_5/(8\cdot 7) = 0,$$

The solution is therefore:

$$y=C (a_{0}\left(1+\frac{x^{3}}{6}+\frac{x^{6}}{180}+\cdots\right)+a_{1}\left(x+\frac{x^{4}}{12}+\frac{x^{7}}{504}+\cdots\right))$$

The ratio test gives $R = \infty$, so each series converges for every $x$. Setting $C_1 = Ca_0$ and $C_2 = Ca_1$, and denoting the two bracketed series by $A(x)$ and $B(x)$, we arrive at the well-known form

{{#content:Q1609}}

== References ==

<references/>

== Further reading ==

* [https://en.wikipedia.org/wiki/Power_series Power series], Wikipedia
* [https://mathworld.wolfram.com/PowerSeries.html Power Series], Wolfram MathWorld
* [https://en.wikipedia.org/wiki/Frobenius_method Frobenius method], Wikipedia
* [https://tutorial.math.lamar.edu/classes/calcii/PowerSeries.aspx Power Series], Paul's Online Math Notes

Power series

2026-09-05T13:23:27Z

RonzzWikiCowriterAI: Proofread: fix grammar ("Redivide", stray commas, "there by", "should converges"), correct the derivative formula (multiply by n!/(n-k)!), fix the perpetuity sum (n from 1, value 2000), fix |x|>1 case to x<−1, complete the dangling binomial "two cases", and repair broken math delimiters in the Airy conclusion. AI-assisted (RonzzWikiCowriter). (via update-page on MediaWiki MCP Server)

'''A power series''' is an infinite sum with the general form

{{#content:Q1667}}

The numbers $a_0$, $a_1$, $a_2$, … are constants, known as the '''coefficients''', and $c$, also a constant, is known as the '''centre''' of the series.

== The simplest form: The geometric series ==

The simplest power series to study is the one in which every coefficient is 1 and the centre is 0:

$$1 + x + x^2 + x^3 + \cdots$$

As seen, each term multiplies the previous one by $x$. This is known as a '''geometric series'''.

For a geometric series, we can calculate the partial sum of the first $N + 1$ terms, $S_N = 1 + x + x^2 + \cdots + x^N$:

Multiply both sides of this equation by $(1 - x)$ and eliminate the brackets:

$$(1-x)S_{N}=(1-x)(1+x+x^{2}+\cdots+x^{N})=(1+x+x^{2}+\cdots+x^{N})-(x+x^{2}+\cdots+x^{N+1})=1-x^{N+1}$$

Dividing both sides again by $(1 - x)$, we therefore have

$$S_{N}=\frac{1-x^{N+1}}{1-x}\qquad(x\neq 1)$$ (1)

When $N \to \infty$, we have

$$\sum_{n=0}^{\infty}x^{n}=\lim_{N \to \infty}S_N=\lim_{N \to \infty}\frac{1-x^{N+1}}{1-x}\qquad(x\neq 1) $$

* '''When $-1 < x < 1$: the series converges.''' As $N$ grows, the number $x^{N+1}$ shrinks towards 0, so the partial sums approach a definite number:

$$\sum_{n=0}^{\infty}x^{n}=\frac{1}{1-x}\qquad\text{whenever }-1<x<1$$

With $x = 0.1$, for instance, the partial sums run 1; 1.1; 1.11; 1.111; …, settling on 1.111…, and indeed $1/(1 - 0.1) = 1/0.9 = 10/9 = 1.111\ldots$

* '''When $x = 1$: the series diverges.''' Notice (1) breaks down here, because $1 - x = 0$ and we cannot divide by 0. The sum is, however, easy to calculate: every term equals 1, so $\lim_{N \to \infty} S_N = \lim_{N \to \infty} (N + 1) = \infty$

* '''Case $x = -1$: the series diverges by oscillation.''' The partial sums run 1, 0, 1, 0, 1, … and never settle on a single number. The formula $1/(1 - x)$ would give $1/2$ at $x = -1$, but notice that is the average of 1 and 0. The formula (1) does not guarantee convergence.

* '''Case $|x| > 1$: the series diverges, as expected.''' When $x > 1$, the infinite sum is the sum of an infinite number of increasingly large positive numbers, which goes to infinity. When $x < -1$, the infinite sum oscillates between positive and negative infinity, as it is dominated by the last term, which may be positive or negative.

=== Real-world application of geometric series ===

If someone promises to pay you €100 each year until the end of your life on earth, how much is that promise worth?

On first thought, this promise is worth quite a lot of money, especially if you count on living for decades more. However, you must take into account inflation, which in Europe averages out to about 5% per year. A payment of €100 one year from now is worth only €$100/1.05$ in today's money, and a payment in two years is worth only €$(100)/(1.05)^2$ in today's money, and so on. A payment of €100 every year, forever, is therefore worth today

$$\sum_{n=1}^{\infty}\frac{100}{1.05^{n}}=2000$$

It is still a handsome sum, if you can live on forever.

== Radius of convergence of a power series ==

In the previous section, we have seen that even the simplest power series, a geometric series, may or may not converge depending on the value of $x$. In general, for a power series centred at $c$, there is a number $R \geq 0$, called the '''radius of convergence''', such that:

* the series converges for every $x$ with $|x - c| < R$ (closer to the centre than $R$);
* the series diverges for every $x$ with $|x - c| > R$.

The set $|x - c| < R$, an interval of length $2R$ centred at $c$, is called the '''interval of convergence of the series'''.

For the two boundary points where $|x - c| = R$, the convergence is uncertain and depends on specific characteristics of a given power series.

In general, consecutive terms of a power series $a_{n+1} (x - c)^{n+1}$ and $a_n (x - c)^n$ have magnitudes in the ratio $|a_{n+1}/a_n| \cdot |x - c|$. If, in the long run, this ratio stays below 1, intuitively the series behaves like a convergent geometric series; if the ratio stays above 1, terms grow larger and larger and the series diverges. Therefore we have

$$R=\lim_{n\to\infty}\left|\frac{a_{n}}{a_{n+1}}\right|\quad\text{when the limit exists}$$

A quick sanity check with geometric series: For the geometric series, every coefficient is 1, so the ratio is 1 and $R = 1$, matching the interval $-1 < x < 1$ found by direct calculation in the section above.

Two special cases exist:

* '''Infinite radius, $R = \infty$:''' the series converges for every $x$. The exponential and sine series in the next section are the standard examples.
* '''Zero radius, $R = 0$:''' the series converges only at the centre itself. For instance the series $1 + x + 2!\,x^2 + 3!\,x^3 + \cdots$ (whose coefficient of $x^n$ is $n!$) has $|a_n/a_{n+1}| = n!/(n + 1)! = 1/(n + 1) \to 0$, so $R = 0$; for any $x \neq 0$ the terms eventually grow without bound.

== The exponential and sine series ==

The power series can be used to approximate known functions that are difficult to calculate directly, such as $e^x$ or $\sin x$.

Imagine we would like to approximate some function $f(x)$ with a power series. We would therefore write

$$f(x)=\sum_{n=0}^{\infty}a_{n}(x-c)^{n}$$

where $c$ and $a_n$ are unknown.

Taking the $k$-th derivative of $f(x)$, we have

$$f^{(k)}(x)=\sum_{n=k}^{\infty}\frac{n!}{(n-k)!}\,a_{n}\,(x-c)^{n-k}$$

Evaluating at $x=c$, we have

$$a_{k}=\frac{f^{(k)}(c)}{k!}$$

Generalising to every $n$, we obtain

{{#content:Q1688}} (2)

=== The exponential series ===

The exponential function $e^x$ has defining properties $\frac{d e^x}{dx}=e^x$ and $e^0 = 1$. Consequently, choosing $c=0$ we have

$$f^{(n)}(0) = 1$$

for every $n$. From (2) we have $a_n = 1/n!$, so

{{#content:Q1684}}

Intuitively, the factorial $n! = 1\cdot 2\cdot 3\cdot\cdots\cdot n$ in the denominator grows much faster than any power function towards infinity, so the terms shrink and the series should converge for every $x$.

Applying the ratio test confirms $R = \infty$:

$$R=\lim_{n\to\infty}\left|\frac{a_{n}}{a_{n+1}}\right|=\lim_{n\to\infty}\frac{(n+1)!}{n!}=\lim_{n\to\infty}(n+1)=\infty$$

A quick sanity check: setting $x = 1$ in the series gives the number $e$ itself: $e = 1 + 1 + 1/2 + 1/6 + 1/24 + \cdots = 2.718\,281\,828\ldots$

=== The sine series ===

Again choosing $c=0$:

{{#content:Q1686}}

Apply once again the ratio test:

$$\lim_{k \to \infty} \frac{\frac{x^{2k+1}}{(2k+1)!}}{\frac{x^{2k+3}}{(2k+3)!}}=\lim_{k \to \infty}\frac{(2k+2)(2k+3)}{x^{2}}= \infty$$

which means the series converges as expected, to $\sin x$.

== The binomial series ==

One more family of power series is useful enough to know by name. For any fixed exponent $p$,

{{#content:Q1689}}

Two cases behave differently and should not be confused:

* '''$p$ a non-negative integer:''' the product $p(p-1)\cdots(p-k+1)$ vanishes once $k$ exceeds $p$, so the series ends after the term in $x^p$ and reduces to the ordinary binomial theorem, which holds for every $x$;
* '''every other $p$:''' the series continues forever and converges for $-1 < x < 1$ as indicated above; at the endpoints $x = \pm 1$ convergence depends on the value of $p$.

== Power series solutions of differential equations ==

A further application of power series is to solve differential equations whose coefficients vary with $x$. Such equations rarely have solutions built from a finite combination of familiar functions, but a solution can sometimes be written down as a power series. Substituting the series into the equation and equating the coefficients of like powers of $x$ turns the differential equation into recurrence relations that fix the coefficients one after another; the constants left free by the recurrence are exactly the arbitrary constants of the equation, and the resulting series solves the equation exactly on its interval of convergence.<ref>{{#cite:Q1577}}</ref><ref>{{#cite:Q1576}}</ref>

=== Case 1: expanding about an ordinary point, the Airy equation ===

The Airy equation

$$y^{\prime\prime} - xy = 0$$ (3)

is named after [[Person:George Biddell Airy|George Biddell Airy]], who employed it in 1838 while studying the intensity of light near a caustic. Its solutions cannot be written as finite combinations of elementary functions, so it is the standard test case for series methods.

Setting

$$y=C\sum_{n=0}^{\infty}a_{n}x^{n}$$

where $a_n$ are unknown coefficients. We have therefore

$$\quad\Rightarrow\quad y^{\prime\prime}=C\sum_{n=0}^{\infty}(n+2)(n+1)\,a_{n+2}\,x^{n},\qquad xy=C\sum_{n=1}^{\infty}a_{n-1}\,x^{n}$$

Substituting these two expressions back into (3) and collecting terms, noting that the equality must hold for every value of $x$, gives

$$\quad a_{2}=0,\qquad (n+2)(n+1)\,a_{n+2}=a_{n-1}\qquad(n\geq 1)$$

We can thereby express all the coefficients in terms of $a_0$ and $a_1$:

$$a_3 = a_0/(3\cdot 2) = a_0/6, \qquad a_6 = a_3/(6\cdot 5) = a_0/180,$$

$$a_4 = a_1/(4\cdot 3) = a_1/12, \qquad a_7 = a_4/(7\cdot 6) = a_1/504,$$

$$a_5 = a_2/(5\cdot 4) = 0, \qquad a_8 = a_5/(8\cdot 7) = 0,$$

The solution is therefore:

$$y=C (a_{0}\left(1+\frac{x^{3}}{6}+\frac{x^{6}}{180}+\cdots\right)+a_{1}\left(x+\frac{x^{4}}{12}+\frac{x^{7}}{504}+\cdots\right))$$

The ratio test gives $R = \infty$, so each series converges for every $x$. Setting $C_1 = Ca_0$ and $C_2 = Ca_1$, and denoting the two bracketed series by $A(x)$ and $B(x)$, we arrive at the well-known form

{{#content:Q1609}}

== References ==

<references/>

== Further reading ==

* [https://en.wikipedia.org/wiki/Power_series Power series], Wikipedia
* [https://mathworld.wolfram.com/PowerSeries.html Power Series], Wolfram MathWorld
* [https://en.wikipedia.org/wiki/Frobenius_method Frobenius method], Wikipedia
* [https://tutorial.math.lamar.edu/classes/calcii/PowerSeries.aspx Power Series], Paul's Online Math Notes

Power series

2026-09-04T15:21:47Z

Qualitative methods

2026-09-04T07:32:23Z

RonzzWikiCowriterAI: New article spun out of Differential equation: equilibria, stability and linearisation, phase-plane analysis of the pendulum, bifurcations, limit cycles and chaos, with full worked demonstrations. AI-assisted (RonzzWikiCowriter). (via create-page on MediaWiki MCP Server)

'''Qualitative methods''', also known as the qualitative theory of differential equations, study differential equations and systems of differential equations without solving them. Instead of formulas for the solutions, they derive the behaviour directly from the equation: the equilibria, their stability, the periodic orbits, and the changes of behaviour (bifurcations) that occur as parameters vary. The approach complements the analytical and numerical methods discussed in the article [[Differential equation]].

The field was founded by [[Person:Henri Poincaré|Henri Poincaré]] in the 1880s, whose geometric methods grew out of the three-body problem of celestial mechanics; [[Person:Aleksandr Lyapunov|Aleksandr Lyapunov]] placed the notion of stability on a rigorous footing in 1892.<ref>{{#cite:Q1578}}</ref>

== Equilibria and stability ==

For an autonomous equation or system d'''y'''/d''t'' = '''f'''('''y'''), an equilibrium (fixed point) is a point '''y'''* at which '''f'''('''y'''*) = 0: a solution placed there never changes. An equilibrium is stable when solutions starting close enough stay close for all time, and asymptotically stable when they are drawn back to it as ''t'' → ∞; otherwise it is unstable.

For a single first-order equation ''y''′ = ''f''(''y''), stability is read from the derivative at the equilibrium: ''f''′(''y''*) < 0 means a small displacement is pushed back, so the equilibrium is stable; ''f''′(''y''*) > 0 means it grows, so the equilibrium is unstable. For instance, ''y''′ = −''y'' has the stable equilibrium 0, while ''y''′ = ''y'' has the unstable equilibrium 0.

For systems the same question is answered by linearising about the equilibrium. Writing '''u''' = '''y''' − '''y'''* and expanding '''f''' to first order gives the linear system

{{#content:Q1677}}

where ''A'' = ''D'''''f'''('''y'''*) is the Jacobian matrix of '''f''' at '''y'''*. In the plane the eigenvalues of ''A'' determine the local phase portrait: two negative real eigenvalues give a stable node; complex eigenvalues with negative real part give a stable spiral; real eigenvalues of opposite signs give a saddle, which is unstable; and purely imaginary eigenvalues give a centre, around which the linearised solutions oscillate. For hyperbolic equilibria, at which no eigenvalue has zero real part, the Hartman–Grobman theorem guarantees that the nonlinear system behaves near the equilibrium exactly like its linearisation.

== The pendulum in the phase plane ==

The undamped pendulum

{{#content:Q1589}}

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

{{#content:Q1617}}

and linearised about each equilibrium,

{{#content:Q1618}}

Near the downward position the linearised equation is the harmonic oscillator, so the phase plane is filled with the closed loops of periodic swinging: a centre. Near the upright position the linearised equation has exponential solutions, so the equilibrium is a saddle: only the two special trajectories that approach along its stable directions ever reach it, and every other nearby state falls away.

== Bifurcations ==

As a parameter changes, the qualitative picture can change abruptly at a bifurcation value: equilibria may appear, merge or vanish. In a saddle-node bifurcation a stable and an unstable equilibrium collide and annihilate each other.

The harvested logistic equation models a population ''N'' that grows logistically and is harvested at the constant rate ''H'':

{{#content:Q1616}}

Its equilibria are found by setting d''N''/d''t'' = 0; the discriminant of the resulting quadratic counts them:

{{#content:Q1678}}

For ''H'' < ''H''c two equilibria coexist, one attracting and one repelling. As ''H'' increases towards ''H''c = ''rK''/4 they move together and merge at the saddle-node bifurcation; beyond ''H''c no equilibrium remains and every population declines to extinction. This is the collapse of an overharvested fishery.

== Periodic orbits and limit cycles ==

In two dimensions a bounded solution eventually approaches an equilibrium or a periodic orbit (Poincaré–Bendixson theorem). Isolated periodic orbits are called limit cycles: nearby trajectories spiral onto them, so the system settles into self-sustained oscillation whatever the initial state. Limit cycles are intrinsically nonlinear, since a linear system oscillates with an amplitude fixed by its initial conditions rather than by the equation itself. The van der Pol oscillator

{{#content:Q1679}}

has, for every μ > 0, an unstable equilibrium at the origin and one stable limit cycle: at small amplitudes the term −μ(1 − ''x''²) feeds energy in, and at large amplitudes it dissipates energy, so all trajectories settle onto the same periodic motion.

== Chaos ==

For systems of three or more equations, bounded solutions need not be periodic: they can be aperiodic yet deterministic, with such sensitivity to initial conditions that nearby states separate exponentially fast. Such motion is called chaotic. [[Person:Edward Lorenz|Edward Lorenz]] found the first example in 1963 in a simple model of atmospheric convection, the system that now bears his name:

{{#content:Q1619}}

For σ = 10, ρ = 28 and β = 8/3 its solutions trace the famous butterfly-shaped chaotic attractor. Because nearby states separate exponentially, the long-term behaviour is unpredictable in practice although the equations are deterministic: this is why weather prediction beyond a limited horizon is impossible. Chaos has since been found throughout physics, biology, chemistry and economics.<ref>{{#cite:Q1578}}</ref>

== References ==

<references/>

== Further reading ==

* [https://en.wikipedia.org/wiki/Qualitative_theory_of_differential_equations Qualitative theory of differential equations], Wikipedia
* [https://en.wikipedia.org/wiki/Phase_plane Phase plane], Wikipedia
* [https://mathshistory.st-andrews.ac.uk/Biographies/Poincare/ Henri Poincaré], MacTutor History of Mathematics

Euler's method

2026-09-04T07:32:22Z

RonzzWikiCowriterAI: New article spun out of Differential equation: derivation, error analysis, convergence demonstration, stability and higher-order Runge-Kutta methods, with a worked code example. AI-assisted (RonzzWikiCowriter). (via create-page on MediaWiki MCP Server)

'''Euler's method''', also called the forward Euler method, is the simplest numerical method for approximating the solution of an initial value problem

:''y''′ = ''f''(''x'', ''y''), ''y''(''x''₀) = ''y''₀.

Given a step size ''h'', it advances an approximate solution along the tangent of the exact solution, one step at a time, producing a polygonal line through the direction field of the equation. The method is named after [[Person:Leonhard Euler|Leonhard Euler]], who described it in his ''Institutiones calculi integralis'' of 1768; it is the prototype of the numerical methods used when no closed-form or open-form solution is available (see [[Differential equation]]).

== The method ==

Choose a step size ''h'' and mark the grid points ''x''ₙ = ''x''₀ + ''nh''. If ''y''(''x'') is the exact solution, Taylor's formula gives the local approximation

{{#content:Q1615}}

Euler's method keeps only the first two terms: the increment over one step is ''h'' times the slope ''f'' evaluated at the beginning of the step. Starting from the initial value and repeating the step,

{{#content:Q1593}}

produces approximate values ''y''ₙ ≈ ''y''(''x''ₙ).

[[File:Euler method.svg|thumb|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).]]

== Error of a single step ==

The step of Euler's method is the first-order Taylor polynomial of the exact solution; the remainder measures how much the step misses. If the exact solution has a continuous second derivative,

{{#content:Q1675}}

for some ξₙ between ''x''ₙ and ''x''ₙ + ''h''. The local truncation error of each step is therefore of order ''h''².

== Global error and convergence ==

A fixed interval of length ''b'' − ''a'' contains (''b'' − ''a'')/''h'' steps, so the local errors accumulate into a total error of order ''h'': halving the step size halves the error. Euler's method is therefore called a first-order method, and the polygonal approximation converges to the exact solution as ''h'' → 0.

For the model problem ''y''′ = ''y'' with ''y''(0) = 1, whose exact solution is ''y'' = ''e''''x'', each Euler step multiplies the estimate by 1 + ''h'':

{{#content:Q1614}}

so the estimate at ''x'' = 1, namely (1 + ''h'')1/''h'', tends to ''e'' as ''h'' → 0. The demonstration is easy to reproduce. A short implementation in Python,

{{#content:Q1680}}

returns about 2.594 after ten steps of size ''h'' = 0.1, already close to the exact value ''e'' = 2.718 281 828…, and the approximation improves as the step size shrinks.

== Stability and higher-order methods ==

Euler's method is not stable for every step size. For the decay equation ''y''′ = −λ''y'' with λ > 0, the numerical values follow ''y''ₙ = (1 − λ''h'')ⁿ''y''₀, which stay bounded only while λ''h'' ≤ 2; for larger steps the computed values grow even though the exact solution decays. Stiff problems therefore demand very small steps or implicit methods.

Better accuracy per step is obtained by evaluating the slope several times inside each step; this is the idea of the '''Runge–Kutta family'''. The midpoint method, a second-order example, first probes the slope at the middle of the interval:

{{#content:Q1676}}

The classical fourth-order Runge–Kutta method, developed by [[Person:Carl Runge|Carl Runge]] and [[Person:Martin Kutta|Martin Kutta]] at the turn of the 20th century, evaluates the slope four times per step and attains an error of order ''h''⁴; it is the default method in most numerical libraries.<ref>{{#cite:Q1576}}</ref><ref>{{#cite:Q1577}}</ref>

== References ==

<references/>

== Further reading ==

* [https://en.wikipedia.org/wiki/Euler_method Euler method], Wikipedia
* [https://mathworld.wolfram.com/EulerForwardMethod.html Euler Forward Method], Wolfram MathWorld
* [https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/ 18.03SC Differential Equations], MIT OpenCourseWare

Laplace transform

2026-09-04T07:32:22Z

RonzzWikiCowriterAI: New article spun out of Differential equation: definition, derivative rule, transform pairs and a full worked demonstration of the method on an initial value problem. AI-assisted (RonzzWikiCowriter). (via create-page on MediaWiki MCP Server)

'''The Laplace transform''' is an integral transform that maps a function ''f''(''t''), defined for ''t'' ≥ 0, to a function ''F''(''s'') of a new variable ''s'', by the formula

{{#content:Q1671}}

For functions of exponential order the integral converges whenever the real part of ''s'' is large enough. The transform is named after [[Person:Pierre-Simon Laplace|Pierre-Simon Laplace]] (1749–1827), who used closely related integral transforms in probability theory in the 1780s; its systematic application to differential equations grew out of the operational calculus developed around the turn of the 20th century.

== Linearity and the derivative rule ==

The Laplace transform is linear: the transform of a linear combination is the same combination of the transforms. Its importance for differential equations rests on the derivative rule, by which differentiation in ''t'' becomes multiplication by ''s'', at the price of subtracting the initial value. For the first derivative the rule follows from one integration by parts:

{{#content:Q1672}}

Applying the rule twice gives the second-derivative rule, written for the function ''y''(''t'') of an initial value problem:

{{#content:Q1620}}

The initial values of the problem therefore appear in the transformed equation directly: an initial value problem for ''y'' becomes an algebraic equation for ''Y''(''s''). The final step, recovering ''y'' from ''Y'', uses a table of standard transforms; the most common pairs are:

{{#content:Q1673}}

== Worked example ==

Consider the initial value problem

:''y''″ + ''y'' = 1, with ''y''(0) = 0 and ''y''′(0) = 0.

'''Step 1, transform the equation.''' Apply the derivative rules to each term, using the given initial values:

{{#content:Q1651}}

'''Step 2, solve for ''Y''(''s'').''' The equation gives ''Y''(''s'') = 1/[''s''(''s''² + 1)]. To invert with the table, expand this rational function in partial fractions with undetermined coefficients:

{{#content:Q1674}}

Comparing coefficients gives ''A'' = 1, ''B'' = −1 and ''C'' = 0, hence

{{#content:Q1652}}

'''Step 3, invert term by term.''' Using the pairs ''L''{1} = 1/''s'' and ''L''{cos ''t''} = ''s''/(''s''² + 1),

{{#content:Q1610}}

The initial conditions entered automatically in step 1, in contrast with the classical method, in which the constants of the general solution are fixed only at the end. The same procedure also handles forcing terms that are discontinuous or impulsive, since the step function and the Dirac delta have simple transforms, whereas classical methods become cumbersome there. Full treatments of the transform method are found in standard textbooks.<ref>{{#cite:Q1576}}</ref><ref>{{#cite:Q1577}}</ref>

== References ==

<references/>

== Further reading ==

* [https://en.wikipedia.org/wiki/Laplace_transform Laplace transform], Wikipedia
* [https://mathworld.wolfram.com/LaplaceTransform.html Laplace Transform], Wolfram MathWorld
* [https://ocw.mit.edu/courses/18-03sc-differential-equations-fall-2011/ 18.03SC Differential Equations], MIT OpenCourseWare

Power series

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