Differential equation

A differential equation is an equation relating an unknown function to its derivatives. It specifies how change depends on the present state, and sometimes on the independent variables themselves. A solution of a differential equation is a function satisfying that relation throughout a stated domain; giving a formula without its domain can conceal a point where the equation is undefined.

The central idea is to turn a local law into a whole trajectory. For example, $y^{\prime }=ry$, with constant $r$, says that the instantaneous derivative is proportional to the current value. If $y(0)=100$ and $r=0.2$ per day, the solution of a differential equation is $y(t)=100e^{0.2t}$, so $y(5)=100e\approx 271.83$. This is an ideal growth model assuming a constant proportional rate; the exponential function emerges from repeatedly applying the same local rule. Changing the sign of $r$ gives exponential decay.

An ordinary differential equation uses one independent variable. A partial differential equation can involve several, such as space and time, and uses partial derivatives. Additional information selects a particular solution of a differential equation: an initial condition describes a starting state, whereas a boundary condition imposes constraints at the edge of a domain. The useful questions extend beyond finding a formula: does a solution of a differential equation exist, is it unique, and what happens to it over time?

An orthogonal trajectory intersects every regular member of a specified family of plane curves at a right angle. If eliminating the family parameter gives its tangent slope $dy/dx=F(x,y)$, the dot product of the two tangent directions gives the orthogonal slope $dy/dx=-1/F(x,y)$ wherever both slopes are finite and nonzero. Vertical and horizontal tangent directions are interpreted geometrically. For the family $y=c{x}^n$ with $n>0$, eliminating $c$ gives $F=ny/x$, and the orthogonal trajectories satisfy $x^2+n{y}^2=C$. For $n=2$ these are ellipses; for $n=1$ they are circles.

A stationary solution of an evolution differential equation is independent of time. Substituting ${\partial }_{t}u=0$ leaves the spatial boundary value problem that determines its profile. For an ordinary differential equation this is an equilibrium; for a partial differential equation it may vary in space. A stationary profile need not have zero flux, although zero flux is forced in a one-dimensional conserved system with reflecting endpoints.

A direction field for $y^{\prime }=f(t,y)$ places a short segment with slope $f(t,y)$ at each sampled point where $f$ is defined. A function graph of a solution of a differential equation follows these segments: its tangent line has exactly the prescribed slope. Segment length is a drawing choice, rather than a speed; arrows, if used, point toward increasing $t$. An isocline collects points with one common slope.

For Newton's law of cooling, $y^{\prime }=-0.1(y-20)$ with time in minutes and temperature in degrees Celsius, the slopes at $y=80,40,20,10$ are respectively $-6,-2,0,1$ degrees Celsius per minute. Each horizontal row is an isocline, because this autonomous differential equation has no explicit time dependence. Above $20$ the solutions of a differential equation descend, below $20$ they rise, and $y=20$ is an equilibrium point.

The highlighted initial condition $y(0)=80$ selects one curve from the family. The picture reveals approach to the equilibrium point before any algebra is done. Under the hypotheses of the Picard-Lindelöf theorem, distinct function graphs of solutions cannot cross or even meet in the time-state plane. A direction field alone does not guarantee uniqueness: several curves can share a tangent line at the same point when those hypotheses fail. See the delayed-start example under initial value problem.

This geometric interpretation is developed in Direction Fields, Isoclines, Integral Curves.

For the first-order ordinary differential equation $y^{\prime }=f(x,y)$ defined on an open subset $D$ of the plane, an isocline of slope $k$ is the level setHere $k$ is a constant. At every point of this isocline, the direction field prescribes a tangent vector proportional to $(1,k)$ for a solution of a differential equation. It does not generally prescribe the tangent line of the isocline itself. The level set can have several components or fail to be a smooth curve, and points outside $D$ are excluded.

For example, take $y^{\prime }=x-y$ on the whole plane. Its isoclines are the parallel straight linesEach line has slope $1$, while the direction field along it has slope $k$. Thus the isocline $y=x$ carries horizontal segments even though the line itself rises. A function graph of a solution of a differential equation follows the segments, rather than following this line. Only the isocline with $k=1$, namely $y=x-1$, is also a solution of a differential equation.^[ref]

The first-order linear differential equation has the exact familyThe initial condition $y(x_0)=y_0$ selects $A=(y_0-x_0+1)e^{x_0}$ and hence a unique solution of a differential equation for every real $x$. The picture shows $y(0)=-1,0,1,2$. For $A=0$ the solution of a differential equation stays on the $k=1$ isocline; for other $A$ it moves through different isoclines. These formulas check the pictured trajectories by direct differentiation, independently of how the direction field was sampled.

The zero-slope isocline $f(x,y)=0$ is called a nullcline. Below the nullcline $y=x$ in this example, $y^{\prime }>0$; above it, $y^{\prime }<0$. For $A>0$, the solution of a differential equation meets it at $x=\log A$, where $y^{\prime }=0$ and $y^{\prime \prime }=1$, a strict local minimum. The nullcline is therefore a useful guide to turning points, although it is not itself a solution of a differential equation. In the equivalent planar autonomous differential equation $\dot{x}=1$, $\dot{y}=x-y$, it is the nullcline for the second component; there is no equilibrium point, because the first component never vanishes.

This makes isoclines useful even when no explicit solution of a differential equation is available: organize a direction field by common slopes, read where function graphs rise or fall, and identify possible turning points on nullclines.

A jump condition relates the one-sided limits of a solution or its derivatives across a discontinuity or singular forcing. Integrating a differential equation across a shrinking interval containing a Dirac delta source determines the permitted jump. A jump condition for an impulse and Rankine-Hugoniot condition are specific instances with different governing equations.

A delay differential equation relates a derivative at the current argument to values at earlier arguments. For example, the Buchstab function obeys $\frac{d}{du}(uw(u))=w(u-1)$ for $u>2$.

The defining feature is memory: a derivative can depend on earlier values of the unknown function. Constant delays, time-dependent delays, and delays depending on the unknown function belong to this wider class. The retarded delay differential equation below isolates a single positive constant delay.^[ref]

The Malthusian delay differential equation is a simple linear member of this class. The method of steps for a delay differential equation constructs its future from a history function of a delay differential equation, and the characteristic equation of a delay differential equation studies its exponential function modes. The exact polynomial steps for a delay equation below make that construction explicit without approximating the solution of a differential equation. Applications include population dynamics and feedback control; the delayed negative feedback model and feedback delay explain why the timing of a return matters as well as its sign.

For a smooth delayed-transfer epidemic model, the incidence $rI(t-\tau )S(t-\tau )$ equals $rIS-r\tau (\dot{I}S+I\dot{S})+O({\tau }^2)$. Substituting the zero-delay equations only inside the correction gives $rIS+\tau rIS\{a+r(I-S)\}+O({\tau }^2)$. The correction has opposite signs in the receiving and departing compartments, preserving their transfer balance.

For nonnegative instantaneous damping $\beta$ and positive delayed gain $\gamma$, characteristic roots obey $\lambda +\beta +\gamma e^{-\lambda }=0$. There is no nonnegative real root. The first imaginary crossing has $\omega \in [\pi /2,\pi )$ and $\beta =-\omega \cot \omega$, $\gamma =\omega /\sin \omega$. In particular stability is retained below this first boundary and $\gamma =\pi /2$ is exact only at $\beta =0$. For small positive $\beta$, the critical gain is $\pi /2+2\beta /\pi +O({\beta }^2)$ and the scaled period tends to four. If the time delay is dimensional $\tau$, that period is multiplied by $\tau$.

This delay differential equation replaces instantaneous logistic inhibition by a population value one delay earlier. A positive history function of a delay differential equation gives positive solutions while they exist. The equilibria are zero and one. Linearization at one gives $x^{\prime }(t)=-x(t-\tau )$, with characteristic equation $\lambda =-e^{-\lambda \tau }$. Delayed sign reversal permits overshoots around one and, at the delayed logistic Hopf threshold, a periodic branch.

A purely imaginary root of $\lambda =-e^{-\lambda \tau }$ has frequency one and first appears at $\tau =\pi /2$. Roots cross with positive real derivative $1/(1+{\tau }_c^2)$; all roots lie in the left half-plane for smaller positive delay, since a hypothetical root $a+ib$ with $a\ge 0$ would require both $|b|\tau \ge \pi /2$ and $|b|\le 1$. A small-amplitude expansion $u=1+a\cos (\omega t)+a^2[\cos (2\omega t)/5+\sin (2\omega t)/10]+\cdots$ yields $\omega =1-3a^2/20+\cdots$ and $\tau =\pi /2+(3\pi -2)a^2/40+\cdots$, verifying a nondegenerate Hopf bifurcation. The limiting period is $2\pi =4{\tau }_c$.

For a single constant delay $\tau >0$, a retarded delay differential equation has the displayed form: delayed values of the unknown function enter, but delayed first derivatives do not. The unknown can also be a vector.^[ref]

Here $t$ denotes time, and the rule applies for $t>0$, with a right derivative at $0$. On $0\le t\le \tau$, the argument $t-\tau$ lies in the prescribed past. Once this interval is known, it supplies the delayed values for $\tau \le t\le 2\tau$. This is the method of steps for a delay differential equation: solve successive ordinary differential equations with a known delayed input, keeping each new endpoint as the next initial condition. For a continuous function $f$ locally Lipschitz continuous in its two state arguments, locally uniformly in $t$, this gives local existence and uniqueness for each continuous history function of a delay differential equation, while the solution of a differential equation remains in the domain of $f$. A nonlinear rule can still produce finite-time blowup.

A history function of a delay differential equation supplies the past interval needed to start a retarded delay differential equation. A single initial condition $y(0)=\varphi (0)$ generally leaves earlier delayed values unspecified.^[ref]

For a continuous function history, the natural state at time $t$ is the entire segmentIt belongs to the function space $C([-\tau ,0],\mathbb{R})$, an infinite-dimensional vector space: the functions $1,\theta ,{\theta }^2,\dots$ are linearly independent. Shifting $y_t$ forward retains the part of the past that the equation can still read. Thus even one scalar unknown can require a state in an infinite-dimensional vector space.

The insufficiency of one value is visible without any approximation. Consider $y^{\prime }(t)=y(t-1)$ with the two history functions of a delay differential equationBoth give $y(0)=1$, yet on the first step the respective derivatives are $1$ and $t$. Direct integration givesIn particular $y_A(1)=2$ while $y_B(1)=3/2$. On the next step,At $t=2$ they are $7/2$ and $8/3$, a gap of $5/6$. These different futures reflect different prescribed histories; each fully specified problem has a unique solution of a differential equation. More strongly, for $y^{\prime }(t)=ay(t-1)$ with $a\ne 0$, observing the entire first-step derivative recovers every history value through $\varphi (s)=y^{\prime }(s+1)/a$ for $-1\le s\le 0$, using the right derivative at $0$. The future can depend on the whole function, rather than a finite list of its samples.

Use the Malthusian delay differential equation with time measured in units of its constant delay:Here $a$ is a real constant. On the first interval $y(t-1)=1$, hence $y(t)=1+at$. On the next interval the delayed input is $1+a(t-1)$, already known, so the method of steps for a delay differential equation givesEach polynomial integrates the preceding interval's delayed polynomial, and its integration constant ensures continuity. At $a=1$, the step endpoints are $y(1)=2$, $y(2)=7/2$, and $y(3)=37/6$. At $t=3/2$, the value is $21/8$, but its derivative is the earlier value $y(1/2)=3/2$. This is the geometric distinction from the ordinary differential equation $y^{\prime }=y$, whose derivative would equal its current value.

More generally, for any integer $n\ge 0$,At an integer join the newly introduced term vanishes. Inside a step, differentiation lowers each power by one and gives exactly $a$ times the preceding step evaluated at $t-1$. Together with $y(0)=1$, this verifies the formula by mathematical induction; it is an exact construction, not a truncated power series.

The history and the new equation need not prescribe matching derivatives at the starting time; their mismatch can reappear later in higher derivatives.^[ref] In this example, $y^{\prime }(0^{-})=0$ and $y^{\prime }(0^{+})=a$. For $a\ne 0$, $y^{\prime }$ is continuous at $t=1$ but $y^{\prime \prime }$ jumps there by $a^2$; at $t=2$, the first jump is in $y^{\prime \prime \prime }$, of size $a^3$. The solution of a differential equation is therefore continuously differentiable for positive time, although it need not have continuous derivatives of every order at step boundaries.

A classical solution is a sufficiently differentiable function that satisfies a differential equation and any prescribed initial or boundary conditions. A weak solution instead satisfies an appropriate integral or distributional identity in a specified function space. For a smooth locally Lipschitz continuous vector field, an ordinary differential equation has a unique local solution for each initial state; this does not by itself ensure global existence or boundedness.

An ordinary differential equation relates an unknown function of one independent variable to its ordinary derivatives. Its order is the highest derivative appearing. Thus $y^{\prime }=-k(y-T_a)$ is first order, while the harmonic oscillator equation $y^{\prime \prime }+{\omega }^{2}y=0$ is second order. A first-order equation in explicit form isHere $t$ need not be time, and $y$ can be a vector, giving a system of coupled ordinary differential equations. A classical solution of a differential equation is a sufficiently differentiable function on an interval which satisfies the equation at every point. For the explicit first-order equation, if $f$ is continuous, this solution of a differential equation has a continuous derivative.

For Newton's law of cooling, the state is a single temperature; for a second-order differential equation, a first-order system also records its first derivative. This is why a starting position alone does not specify a harmonic oscillator trajectory: its starting velocity matters too.

The power of an ordinary differential equation is that one rule describes a whole family of histories. An initial value problem chooses one history when uniqueness holds. An autonomous differential equation has $f$ independent of $t$; its equilibrium points satisfy $f(y_{\ast })=0$ and give constant solutions of a differential equation. Equations with explicit time dependence can instead respond to a changing environment.

An autonomous differential equation for $f(\eta )$ can be reduced by one order on a monotone solution branch by using $f$ as the new independent variable. Setting $G(f)=df/d\eta$ and applying the chain rule gives the displayed derivative identities. The absence of explicit $\eta$ dependence makes the resulting equation involve only $f,G$ and derivatives of $G$. A point with $G=0$ is singular for this coordinate change; for example $f\sim b{\eta }^2/2$ gives $G\sim \sqrt{2bf}$ on an appropriate branch, so $G_f$ need not remain finite there.

Where $y\ne 0$, its logarithmic derivative $q=y^{\prime }/y$ changes this nonlinear second-order ordinary differential equation into the first-order linear differential equation $q^{\prime }=a(x)q$. For continuous $a$, initial values $y(x_0)=y_0\ne 0$ and $y^{\prime }(x_0)=v_0$ therefore giveDifferentiation proves both the equation and the initial data. The exponential preserves the sign of $y_0$, so the solution never reaches the singular value zero on any finite interval where $a$ is continuous. For $a(x)=2x$ and $y_0=v_0=1$, the expression reduces to $y(x)=\exp ({\int }_0^x{e}^{t^2}dt)$.

A system of ordinary differential equations specifies derivatives of several unknown functions of one independent variable. A vector-valued form ${\mathbf{y}}^{\prime }=\mathbf{f}(t,\mathbf{y})$ bundles its scalar equations. A scalar second-order ordinary differential equation $y^{\prime \prime }=F(t,y,y^{\prime })$ becomes the first-order system $(y_1^{\prime },y_2^{\prime })=(y_2,F(t,y_1,y_2))$ by introducing position and velocity variables. Conversely the first equation identifies $y_2=y_1^{\prime }$ and the second recovers the original equation. Its initial condition must specify both variables.

An implicit ordinary differential equation relates the derivative to the independent and dependent variables without requiring it to be explicitly isolated. For example $y{y}^{\prime }=1$ is meaningful at every pair $(x,y)$ as an equation, but no differentiable solution of a differential equation can pass through $y=0$: its left side would vanish there. Dividing by $y$ gives $y^{\prime }=1/y$ only on the domain $y\ne 0$. Conversely multiplying an explicit equation by a vanishing denominator can introduce points where the original equation was undefined. A solution of the multiplied equation at such a point need not solve the original initial value problem.

A constant function $y(x)=c$ is an equilibrium solution of $y^{\prime }=F(x,y)$ on an interval when $F(x,c)=0$ throughout that interval. Dividing by a factor such as $y-c$ during separation of variables excludes this solution, so it must be checked separately. For an autonomous differential equation, the constant state is an equilibrium point.

A first-order ordinary differential equation contains the first derivative of its unknown function and no higher derivative. When solved for that derivative it has the displayed form. With an initial condition, local existence and uniqueness follow under appropriate regularity assumptions on $F$; the order alone does not guarantee either property.

On an interval with $x\ne 0$, the substitution $y=xv$ and the product rule give the displayed identity. An equation $x{y}^{\prime }-y=g(y/x)$ becomes $v^{\prime }=g(v)/x^2$ and permits separation of variables wherever $g(v)\ne 0$. Roots $g(v_0)=0$ also give solutions $y=v_{0}x$ and must be retained if division by $g$ was used. This substitution works for this displayed class even though its explicit factor $1/x$ means that $y^{\prime }=y/x+g(y/x)/x$ need not be a homogeneous equation of the usual form $y^{\prime }=F(y/x)$.

On a branch where a differentiable function $y(x)$ has a nonzero derivative, the inverse function theorem allows the ordinary differential equation to be rewritten for $x(y)$. A nonlinear equation in one direction can become a linear ordinary differential equation in the other. For example, $y^{\prime }=y/(x+y+c)$ becomesThe integrating factor $1/y$ gives $(x/y{)}^{\prime }=1/y+c/y^2$, hence $x+c=y(\ln |y|+C)$. This family must retain the original denominator restriction $x+y+c\ne 0$. The separate zero solution $y=0$ on intervals avoiding $x=-c$ is lost by the inversion and must be restored.

For a twice differentiable solution, put $p=y^{\prime }$. Differentiation gives $[x+f^{\prime }(p)]p^{\prime }=0$. One family consists of straight lines $y=Cx+f(C)$ with constant slope $C$. The other consists of the envelope of those lines, given parametrically by $x=-f^{\prime }(p)$ and $y=f(p)-p{f}^{\prime }(p)$ where these equations define a differentiable graph with derivative $p$. An envelope is a singular solution not obtained by fixing $C$. If only continuous first derivatives are required, a solution may also join a line to its envelope at their point of tangency, provided the joined function satisfies the original equation. Such joins need not have a continuous second derivative.

A homogeneous first-order differential equation has the form $y^{\prime }=F(y/x)$ on a domain where $x\ne 0$. The substitution $v=y/x$, so $y^{\prime }=v+x{v}^{\prime }$, reduces it toWhere $F(v)-v\ne 0$, this is a separable differential equation, with $\int dv/[F(v)-v]=\ln |x|+C$. Values $v_0$ satisfying $F(v_0)=v_0$ give the additional straight-line solutions $y=v_{0}x$, which must be checked before division. This scaling-based use of “homogeneous” is different from a homogeneous linear differential equation.

For example, $y^{\prime }=y/x+x/y$ gives $F(v)=v+1/v$, so $v{v}^{\prime }=1/x$ and $y^2=x^2(2\ln |x|+C)$. Real explicit branches are $y=\pm x\sqrt{2\ln |x|+C}$ on intervals where the expression under the square root is positive. Zeros of a denominator and endpoints of a branch must still be checked in the original ordinary differential equation.

For real $a\ne 0$ and $x\ne 0$, set $u=y/x$ to obtain $x{u}^{\prime }=a\tan u$. On nonconstant branches, separation gives $\log |\sin u|=a\log |x|+C$, hence $\sin (y/x)=C_1|x{|}^a$. Zeros of the divided-out tangent give $y=k\pi x$, already represented by $C_1=0$. Points where $\cos (y/x)=0$ remain excluded because the original differential equation is undefined. An implicit trigonometric identity alone does not specify a connected branch or its domain.

For $x>0$ on a domain where the tangent is finite, put $v=y/x$. The homogeneous first-order differential equation becomes $x{v}^{\prime }=-\tan v$. Direct differentiation gives $(x\sin v{)}^{\prime }=0$, hence $x\sin (y/x)=C$, without dividing by $\sin v$ and losing constant-ratio solutions. If $C=x_0>0$ and $y/x\to \pi /2$ as $x\downarrow x_0$, two branches are $y=x\arcsin (x_0/x)$ and $y=x[\pi -\arcsin (x_0/x)]$, defined for $x>x_0$. They approach the same finite endpoint but have unbounded derivatives. The original tangent is undefined at the endpoint, so neither is a classical solution through that point. An implicit relation extending continuously to an initial point does not by itself solve an initial value problem there.

An implicit homogeneous first-order differential equation $y/x=F(p)$, where $p=dy/dx$, can be reduced by treating the slope as an auxiliary variable. Differentiating $y=xF(p)$ givesWhere the divisions are valid, this is a separable differential equation. Constant slopes lost when dividing by $p-F(p)$ must be checked in the original equation.