# Ordinary differential equation
## MicroSim spec
- **Recommended sim type:** [[Phase_space|phase space]] / [[Attractor|attractor]]
- **Microsimmability score:** 82/100
- **Layout:** drawing region (canvas) on top; control region (sliders/buttons) below.
### Parameters (tunable controls)
- `Rate coefficient`
- `Forcing`
- `Initial value`
### What animates
A solution curve grows, decays, or oscillates as the equation's coefficients change.
### Learning objective
Relate an ODE's coefficients to the shape of its solution over time.
## Links (Wikipedia order)
<!-- injected from _registry/childlinks/Ordinary_differential_equation.json (2026-07-30T02:09:12Z) -->
`Abel's_identity` · `Abelian_integral` · `Alexander_Hrennikoff` · `Alexis_Clairaut` · `Alfred_Clebsch` · `Algebraic_analysis` · `American_Mathematical_Society` · `Analytical_mechanics` · `Andrei_Polyanin` · `Ansatz` · `Antiderivative` · `Applied_mathematics` · `Arthur_Cayley` · `Artistic_License` · `Astronomy` · `Asymptotic_expansion` · `Augustin-Louis_Cauchy` · `Autonomous_system_(mathematics)` · `Bernoulli_family` · `Biology` · `Boundary_value_problem` · `Bäcklund_transform` · `COPASI` · [[Calculus]] · [[Calculus_of_variations]] · `Cannon` · `Carathéodory's_existence_theorem` · `Carl_Neumann` · `Cartesian_product` · `Cauchy_problem` · `Cauchy–Kovalevskaya_theorem` · [[Chaos_theory]] · `Chebfun` · [[Chemistry]] · [[Complex_analysis]] · `Complex_number` · `Computer_algebra_system` · `Constant_of_integration` · `Continuous_function` · `Continuum_mechanics` · `Crank–Nicolson_method` · `David_Halliday_(physicist)` · `Degree_of_a_polynomial` · `Delay_differential_equation` · `Dependent_and_independent_variables` · `Derivative` · `Differentiable_function` · `Differential-algebraic_system_of_equations` · [[Differential_equation]] · `Differential_geometry` · `Differential_operator` · `Dirac_delta_function` · `Dirichlet_boundary_condition` · [[Ecology]] · `Economics` · `Edward_Lindsay_Ince` · `Eigenfunction` · `Elementary_function` · `Encyclopedia_of_Mathematics` · [[Engineering]] · `Equation` · `Ernst_Leonard_Lindelöf` · `Erwin_Kreyszig` · `Euler's_formula` · `Euler_method` · `Exact_differential_equation` · `Exponential_response_formula` · `Exponential_stability` · `Felice_Casorati_(mathematician)` · `Ferdinand_Georg_Frobenius` · `Finite_difference_method` · `Finite_element_method` · `Finite_volume_method` · `Floquet_theory` · [[Fourier_analysis]] · `Frobenius_method` · `Function_(mathematics)` · [[Functional_analysis]] · `Fundamental_theorem_of_calculus` · `GNU_Octave` · `Galerkin_method` · `Geology` · [[Geometry]] · [[George_Boole]] · `George_F._Simmons` · `George_Green_(mathematician)` · `Gerald_Teschl` · `Gottfried_Wilhelm_Leibniz` · `Gradient` · `Green's_function` · `Group_theory` · `Grönwall's_inequality` · `Harmonic_analysis` · `Harold_Jeffreys` · `Hessian_matrix` · `Holonomic_function` · `Homogeneous_differential_equation` · `Hypercomplex_analysis` · `Inexact_differential_equation` · `Infinite_element_method` · `Infinitesimal_transformation` · `Infinity` · `Initial_value_problem` · `Integral` · `Integral_curve` · [[Integral_transform]] · `Integrating_factor` · `Integration_by_substitution` · `Integro-differential_equation` · [[Isaac_Newton]] · `Jacob_Bernoulli` · `Jacques_Charles_François_Sturm` · `Jean_Gaston_Darboux` · `Jean_Le_Rond_d'Alembert` · `Jet_bundle` · `John_Crank` · `Joseph-Louis_Lagrange` · `Joseph_Fourier` · `Joseph_Liouville` · `Julia_(programming_language)` · `Józef_Maria_Hoene-Wroński` · `Lamar_University` · `Laplace_transform_applied_to_differential_equations` · `Lax_pair` · `Lazarus_Fuchs` · [[Least-squares_spectral_analysis]] · `Leibniz's_notation` · `Leonhard_Euler` · [[Lie_group]] · `Lie_theory` · `Limit_(mathematics)` · `Linear_combination` · `Linear_differential_equation` · `List_of_dynamical_systems_and_differential_equations_topics` · `List_of_linear_ordinary_differential_equations` · `List_of_named_differential_equations` · `List_of_nonlinear_ordinary_differential_equations` · `List_of_nonlinear_partial_differential_equations` · `Lists_of_integrals` · [[Lyapunov_stability]] · `MATLAB` · `Macmillan_Publishers` · `Maple_(software)` · `Martin_Kutta` · `Mathematical_analysis` · `Mathematics` · `Matrix_calculus` · `Matrix_differential_equation` · `Maxima_(software)` · `Measure_(mathematics)` · `Meteorology` · `Method_of_characteristics` · `Method_of_undetermined_coefficients` · `Microlocal_analysis` · `Nail_H._Ibragimov` · `Natural_science` · `Neumann_boundary_condition` · [[Nonlinear_system]] · `Norman_Levinson` · `Notation_for_differentiation` · [[Numerical_integration]] · `Numerical_methods_for_ordinary_differential_equations` · `P-adic_analysis` · `P-adic_number` · [[Partial_differential_equation]] · `Peano_existence_theorem` · `Periodic_boundary_conditions` · `Perturbation_theory` · `Peter_Gustav_Lejeune_Dirichlet` · `Petrov–Galerkin_method` · `Phase_portrait` · [[Phase_space]] · `Philip_Hartman` · `Phyllis_Nicolson` · [[Physics]] · `Picard–Lindelöf_theorem` · `Pierre-Simon_Laplace` · `Polynomial` · [[Population_dynamics]] · `Power_series_solution_of_differential_equations` · `Projectile` · `Projectile_motion` · `Providence,_Rhode_Island` · `Quadrature_(mathematics)` · `Quaternionic_analysis` · `Rate_of_convergence` · `Real_analysis` · `Real_number` · [[Recurrence_relation]] · `Reduction_of_order` · `Representation_theory` · `Restriction_(mathematics)` · `Riccati_equation` · `Richard_Courant` · `Robert_Resnick` · `Robin_boundary_condition` · `Rudolf_Lipschitz` · `Runge–Kutta_methods` · `SageMath` · `SciPy` · `Scilab` · `Separation_of_variables` · `Series_(mathematics)` · `Singular_matrix` · `Singular_solution` · `Social_science` · `Society_for_Industrial_and_Applied_Mathematics` · `Sofya_Kovalevskaya` · `Sophus_Lie` · `Special_functions` · `Stack_Exchange` · `Stochastic_differential_equation` · `Stochastic_partial_differential_equation` · `Sturm–Liouville_theory` · `Summation` · `SymPy` · `System_of_differential_equations` · `System_of_linear_equations` · `Taylor_series` · `Trajectory` · `Tropical_analysis` · `Variable_(mathematics)` · `Variation_of_parameters` · `Vector-valued_function` · `Vector_calculus` · `Victor_Gustave_Robin` · `WKB_approximation` · `Well-posed_problem` · `Witold_Hurewicz` · `Wronskian` · `Émile_Picard`
## From the Real GENERATIVE library (beauty pass)
*Ordinary differential equation — image hotlinked from Wikimedia Commons (via the Real G.E.N.E.R.A.T.I.V.E. course library, STEM and Music room). [Details & license](https://commons.wikimedia.org/wiki/File:Parabolic_trajectory.svg).*
> In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable. As with other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions.[1] The term "ordinary" is used in contrast with partial differential equations (PDEs) which may be with respect to more than one independent variable,[2] and, less commonly, in contrast with stochastic differential equations (S ([Wikipedia](https://en.wikipedia.org/wiki/Ordinary_differential_equation))
<!-- BEAUTY-PASS-MEDIA:END -->
> p5.js MicroSim stub · part of Systems Thinking · [Wikipedia source](https://en.wikipedia.org/wiki/Ordinary_differential_equation)
> Relation: subfield of systems thinking.
## Concept summary
In mathematics, an ordinary differential equation (ODE) is a [[Differential_equation|differential equation]] (DE) dependent on only a single independent variable. As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions.
## Build checklist
- [ ] Claim it: set `status: in-progress` + `lease` + `leased_at`
- [ ] Write the child page explaining the concept (tie it back to systems thinking)
- [ ] Finalize parameter ranges and defaults
- [ ] Implement the p5.js sketch (drawing + control regions)
- [ ] Add caption + the learning objective on the page
- [ ] Set `status: done`
## p5.js sketch
```javascript
// MicroSim: Ordinary differential equation
let controls = {};
function setup() {
// createCanvas(...); create sliders for the parameters above.
}
function draw() {
// background(...); read controls; render phase space / attractor; respond live.
}
```
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Back to Systems Thinking · Wikipedia: [Ordinary differential equation](https://en.wikipedia.org/wiki/Ordinary_differential_equation)
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## Semiotic profile
> *The semiotic universals this article invokes, machine-derived from the crossref — **unverified** (born so). Populated 2026-07-06 for the Systems room.*
**Universals (4):** 🟡 system (13) · 🟡 science (5) · 🟢 attractor chaos (4) · 🟢 oscillation (1)
**Enter by sign:** Systems Semiotic Gateway · Alphabetum · Icon Registry · ← Systems Portal
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## Wikipedia : Wikitube
**Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Ordinary_differential_equation) : [Wikitube](https://en.wikitube.io/wiki/Ordinary_differential_equation)
## Previous hub tags
Tree parents: [[Complex_system]] · [[Cybernetics]] · [[Decision_theory]] · [[Dynamical_system]] · [[Emergence]] · [[Feedback]] · [[Phase_space]] · [[Self-organization]] · [[Systems_science]] · [[Systems_theory]].
Legacy hubs: none.
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*Sources: 2 legacy notes. Minted wave 1, 2026-07-30 (v1.6 order).*