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This second-order ordinary differential equation has the independent entire Airy functions $\mathrm{Ai}$ and $\mathrm{Bi}$ as a standard basis. It is distinct from the dispersive PDE in Airy equation. Its general solution is $a_1\mathrm{Ai}(Z)+a_2\mathrm{Bi}(Z)$.

For $c(y)=1-y^2$ in the strip $|y|<1$, the travel-time coordinate is $s(y)=\mathrm{artanh}y$. The characteristic curves areequivalently $y=\tanh (C\mp x)$. Outside the strip the same separated ordinary differential equation gives hyperbolic cotangent branches, and the equilibrium curves $y=\pm 1$ are also characteristic.

A characteristic speed is an eigenvalue of the flux Jacobian. Along a corresponding characteristic curve, an associated wave variable obeys an ordinary differential equation obtained by projecting the governing system onto the matching left eigenvector.

Given distinct nodes $c_i$, a collocation Runge-Kutta method uses the Lagrange basis ${\ell }_j$ with $a_{ij}={\int }_0^{c_i}{\ell }_j(\tau )d\tau$ and $b_j={\int }_0^1{\ell }_j(\tau )d\tau$. Its stages enforce the differential equation at the collocation nodes.

A smooth vector field is complete if its maximal integral curves of a vector field exist for every real time. Its local flow is then a global one-parameter group of diffeomorphisms; uniqueness of ordinary differential equations proves the group law and the inverse-time identity. On a compact manifold without boundary, every smooth vector field is complete: compactness gives a uniform positive local existence interval through every point, so a curve can be successively extended past any proposed finite endpoint. On $\mathbb{R}$, $x^2{\partial }_x$ is not complete because its solution $x(t)=x_0/(1-t{x}_0)$ has finite-time blowup for $x_0>0$.

Control theory studies dynamical input-output systems, feedback, stability, and response.

For a positive-order constant-coefficient ordinary differential equation, the kernel of multiplication by a coordinate gives $xP(D)u=0$ exactly when $P(D)u=C{\delta }_0$. Hence every solution is $u=v+CE$, where $P(D)v=0$ and $E$ is any fundamental solution of a linear differential operator. Taking a retarded fundamental solution of a constant-coefficient ordinary differential operator shows that the extra freedom is a single jump in the derivative of order one below the operator order. The coordinate multiplies the already differentiated distribution: this is a different operator from $P(D)(xu)$.

For $a>0$, the positive amplitude ordinary differential equation $R^{\prime }=aR-R^3$ has solution $R(t)=\sqrt{a}[1+(a/R_0^2-1)e^{-2at}{]}^{-1/2}$. This follows from the logistic differential equation for $R^2$. Every $R_0>0$ approaches $\sqrt{a}$ as a monotone function, and the positive distance to this limit decreases; an oscillating complex wave carrying this amplitude has no ordinary monotone ordering.

A real vector field $f$ is dissipative in an inner product iffor all $x,y$. Two solutions of the ordinary differential equation $y^{\prime }=f(y)$ satisfyThus their distance never increases. The linear case $f(y)=Ay$ recovers the dissipative operator condition. B-stability asks whether a Runge-Kutta method preserves this contraction for every positive step size.

A solution of an ordinary differential equation has finite-time blow-up when its maximal interval of existence has a finite endpoint and its norm becomes unbounded on approach to that endpoint.

A nonnegative fluid model drains in finite time if all its coordinates approach zero at a finite terminal time. A ratio-based ordinary differential equation can become undefined at that endpoint; an absorbing extension is an additional modeling convention.

When backlog $N$ and attempt-control scale $S$ are large with $N/S\approx \kappa$, idle and success probabilities approach $e^{-\kappa }$ and $\kappa e^{-\kappa }$. Expected update increments then motivate a fluid model described by ordinary differential equations.

If $F\in C^1(\mathbb{R}\times {\mathbb{R}}^d)$ and $|F(t,x)|\le K(1+|x|)$, every characteristic curve is defined for all real time. The Gronwall inequality bounds $1+|X(t)|$ on each bounded time interval, so local ordinary differential equation existence cannot terminate through escape to infinity. Uniqueness and the variational equation give a global $C^1$ spatial flow diffeomorphism, with Jacobian $J>0$ satisfying $J_t=(\mathrm{div}F)(t,X)J$.

A gravity-current box model replaces the current by a well-mixed region with uniform depth and density. Integral volume, scalar, and front-speed balances then reduce the spreading problem to ordinary differential equations.

A solution specified by a relation $F(x,y)=C$, instead of by an explicit graph $y=f(x)$. Differentiating the relation verifies its ordinary differential equation. Where $F_y\ne 0$, the implicit function theorem supplies a local graph; other points may require a different parameterization.

An initial value problem specifies an ordinary differential equation together with the state at a starting time. For a continuous linear differential equation $u^{\prime }=A(t)u$, every prescribed finite vector $u_0$ at $t_0$ determines a unique solution on the coefficient interval. This is different from a boundary value problem, which can prescribe values at distinct endpoints.

An integral curve follows a vector field by solving the displayed ordinary differential equation. For a smooth vector field, integral curves through initial points assemble into its local flow.

For a continuous Loewner driving function $U$, $T_z$ is the maximal lifetime of $g_t(z)$ in the complex upper half-plane under the Chordal Loewner equation. A finite lifetime has $\mathrm{Im}{g}_t(z)\to 0$ as $t\uparrow T_z$: otherwise the continuous driver and the ordinary differential equation continue the solution. A swallowed point need not have been visited by the Loewner trace; it may lie in a disconnected pocket of the filled hull.

The $n$th jet space for a scalar function $u(x)$ has local coordinatesIt records the value of a function and its derivatives through order $n$ at one point, allowing an order-$n$ ordinary differential equation to be treated as a hypersurface and a point transformation to act through its prolongation of a vector field.

A spatially constant blowup profile for a wave equation solves an ordinary differential equation. Choose smooth compactly supported initial data which agree with the profile on a ball larger than its blowup time times the wave speed. Finite propagation speed preserves agreement in an interior light cone, forcing finite-time breakdown of the localized solution. For $u_{tt}-\Delta u=u^3$, the profile $u(t)=\sqrt{2}/(1-t)$ is one example.