# Differential equation
## MicroSim spec
### Parameters (tunable controls)
- `Rate k` · -2–2 · coefficient in dy/dt = k*y (decay vs. growth)
- `Initial value y0` · -10–10 · starting state of the solution
- `Time step dt` · 0.01–0.5 · integration step (shows numerical stability)
### What animates
The solution curve y(t) is integrated and drawn in real time as parameters change.
### Learning objective
Show how a first-order ODE's coefficient and step size shape the solution curve.
## Links (Wikipedia order)
<!-- injected from _registry/childlinks/Differential_equation.json (2026-07-30T02:09:12Z) -->
`Abel's_identity` · `Abstract_algebra` · `Abstract_differential_equation` · `Acta_Eruditorum` · `Affine_geometry` · `Alexander_Hrennikoff` · `Alexis_Clairaut` · `Algebra` · `Algebraic_analysis` · `Algebraic_geometry` · `Algebraic_number_theory` · `Algebraic_topology` · `American_Journal_of_Physics` · `American_Mathematical_Society` · `Analytic_geometry` · `Analytic_number_theory` · `Ansatz` · `Antiderivative` · `Applied_mathematics` · `Arithmetic` · `Arithmetic_geometry` · `Astronomy` · `Asymptotic_expansion` · `Augustin-Louis_Cauchy` · `Autonomous_system_(mathematics)` · `Bernoulli_differential_equation` · `Bill_Schelter` · `Biology` · `Boundary_value_problem` · [[Calculus]] · [[Calculus_of_variations]] · `Carathéodory's_existence_theorem` · `Carl_Neumann` · `Category_theory` · `Cauchy_problem` · `Cauchy–Kovalevskaya_theorem` · [[Chaos_theory]] · [[Chemistry]] · `Classical_mechanics` · `Closed-form_expression` · `Combinatorics` · `Commutative_algebra` · [[Complex_analysis]] · `Complex_geometry` · [[Complex_system]] · `Computational_complexity_theory` · `Computational_geometry` · `Computational_mathematics` · `Computer_algebra` · `Computer_algebra_system` · [[Computer_science]] · `Continuous_function` · `Continuum_mechanics` · [[Control_theory]] · `Convex_geometry` · `Crank–Nicolson_method` · `Daniel_Bernoulli` · `Delay_differential_equation` · `Dependent_and_independent_variables` · `Derivative` · `Differential-algebraic_system_of_equations` · `Differential_geometry` · `Differential_operator` · `Differential_topology` · [[Diffusion]] · `Diophantine_geometry` · `Dirac_delta_function` · `Dirichlet_boundary_condition` · `Discrete_geometry` · `Discrete_mathematics` · [[Dynamical_systems_theory]] · `Earl_A._Coddington` · `Economics` · `Elasticity_(physics)` · `Electrostatics` · `Elementary_algebra` · [[Engineering]] · [[Engineering_mathematics]] · `Equations_of_motion` · `Ergodic_theory` · `Ernst_Leonard_Lindelöf` · `Euclidean_geometry` · `Euler_method` · `Euler–Lagrange_equation` · `Exact_differential_equation` · `Exponential_response_formula` · `Exponential_stability` · `Field_(mathematics)` · `Finite_difference_method` · `Finite_element_method` · `Finite_geometry` · `Finite_volume_method` · `Floquet_theory` · `Foundations_of_mathematics` · [[Fourier_analysis]] · `Function_(mathematics)` · `Function_of_a_real_variable` · [[Functional_analysis]] · `Functional_differential_equation` · `Functional_equation` · `Fundamental_theorem_of_calculus` · `Future_of_mathematics` · `Galerkin_method` · [[Game_theory]] · `General_topology` · `Geology` · `Geometric_topology` · [[Geometry]] · [[George_Boole]] · `George_Green_(mathematician)` · `Gerald_Teschl` · `Glossary_of_mathematical_symbols` · `Gottfried_Wilhelm_Leibniz` · [[Graph_theory]] · `Green's_function` · `Group_theory` · `Harmonic_analysis` · `Harmonic_oscillator` · `Harold_Jeffreys` · `Heat` · `Heat_equation` · [[Heat_transfer]] · `History_of_calculus` · `History_of_mathematics` · `Holonomic_function` · `Homogeneous_differential_equation` · `Homological_algebra` · `Homotopy_theory` · `Hypercomplex_analysis` · `Infinite_element_method` · `Infinity` · `Informal_mathematics` · `Information_geometry` · [[Information_theory]] · `Initial_condition` · `Initial_value_problem` · `Integral` · `Integral_equation` · [[Integral_transform]] · `Integrating_factor` · `Integration_by_substitution` · `Integro-differential_equation` · [[Isaac_Newton]] · `Jacob_Bernoulli` · `Jacques_Charles_François_Sturm` · `Jean_Le_Rond_d'Alembert` · `Jet_bundle` · `John_Crank` · `Joseph-Louis_Lagrange` · `Joseph_Fourier` · `Joseph_Liouville` · `Józef_Maria_Hoene-Wroński` · `Korteweg–De_Vries_equation` · `Lagrangian_mechanics` · `Lamar_University` · `Large_numbers` · [[Least-squares_spectral_analysis]] · `Leonhard_Euler` · `Limit_(mathematics)` · [[Linear_algebra]] · `Linear_differential_equation` · `Linear_equation` · `Linearization` · `List_of_films_about_mathematicians` · `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` · `Lists_of_mathematics_topics` · [[Lyapunov_stability]] · `Maple_(software)` · `Martin_Kutta` · `MathWorld` · `Mathematical_analysis` · `Mathematical_and_theoretical_biology` · `Mathematical_chemistry` · `Mathematical_economics` · `Mathematical_finance` · `Mathematical_linguistics` · `Mathematical_logic` · [[Mathematical_model]] · [[Mathematical_optimization]] · `Mathematical_physics` · `Mathematical_psychology` · `Mathematical_sociology` · `Mathematical_statistics` · `Mathematics` · `Mathematics_and_art` · `Mathematics_education` · `Matrix_calculus` · `Maxima_(software)` · `Measure_(mathematics)` · `Mechanics` · `Method_of_Fluxions` · `Method_of_characteristics` · `Method_of_undetermined_coefficients` · `Microlocal_analysis` · `Multivariable_calculus` · `Musical_instrument` · `Natural_science` · `Navier–Stokes_existence_and_smoothness` · `Neumann_boundary_condition` · `Newton's_law_of_cooling` · [[Newton's_laws_of_motion]] · [[Nonlinear_system]] · `Norman_Levinson` · `Notation_for_differentiation` · `Number` · `Number_theory` · `Numerical_analysis` · [[Numerical_integration]] · `Numerical_methods_for_ordinary_differential_equations` · `Numerical_methods_for_partial_differential_equations` · [[Operations_research]] · [[Ordinary_differential_equation]] · `P-adic_analysis` · `P-adic_number` · [[Partial_differential_equation]] · `Peano_existence_theorem` · [[Pendulum]] · `Periodic_boundary_conditions` · `Perturbation_theory` · `Peter_Gustav_Lejeune_Dirichlet` · `Petrov–Galerkin_method` · `Phase_portrait` · [[Phase_space]] · `Philosophy_of_mathematics` · `Phyllis_Nicolson` · [[Physics]] · `Picard–Lindelöf_theorem` · `Pierre-Simon_Laplace` · [[Population_dynamics]] · `Power_series_solution_of_differential_equations` · `Probability_theory` · [[Projective_geometry]] · `Pure_mathematics` · `Quantum_field_theory` · [[Quantum_mechanics]] · `Quaternionic_analysis` · [[Radioactive_decay]] · `Randomness` · `Rate_of_convergence` · `Real_analysis` · `Recreational_mathematics` · [[Recurrence_relation]] · `Reduction_of_order` · `Representation_theory` · `Richard_Courant` · `Richard_DiPrima` · `Robin_boundary_condition` · `Rudolf_Lipschitz` · `Runge–Kutta_methods` · `SageMath` · `Scientific_law` · `Separation_of_variables` · `Series_(mathematics)` · [[Set_theory]] · `Social_science` · `Sofya_Kovalevskaya` · `Sound` · `SourceForge` · `Special_functions` · `Statistical_mechanics` · `Statistics` · `Stochastic_differential_equation` · `Stochastic_partial_differential_equation` · `Stochastic_process` · `Sturm–Liouville_theory` · `SymPy` · `Symmetry` · `System_of_differential_equations` · [[Systems_science]] · `Theory_of_computation` · `Thin-film_equation` · `Timeline_of_mathematics` · `Topology` · `Tropical_analysis` · `Type_theory` · `Ultrametric_space` · `Variable_(mathematics)` · `Variation_of_parameters` · `Vector_calculus` · `Victor_Gustave_Robin` · `WKB_approximation` · [[Wave_equation]] · [[Wayback_Machine]] · `Well-posed_problem` · `Wiener_process` · `Wikisource` · `Wolfram_Mathematica` · `Wronskian` · `Xcas` · `Émile_Picard`
## From the Real GENERATIVE library
*Differential equation — placed from the Real G.E.N.E.R.A.T.I.V.E. course library (STEM and Music room). Source: Wikimedia Commons (via Wikipedia article media). [Details & license](https://commons.wikimedia.org/wiki/File:Wikiquote-logo.svg).*
> In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives.[1] In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differe ([Wikipedia](https://en.wikipedia.org/wiki/Differential_equation))
<!-- REAL-GENERATIVE-MEDIA:END -->
> p5.js MicroSim stub · part of Systems Engineering · [Wikipedia source](https://en.wikipedia.org/wiki/Differential_equation)
## Concept summary
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.
## Build checklist
- [ ] Claim it: set `status: in-progress` + `lease` + `leased_at`
- [ ] Write the child page explaining the concept (tie it back to Systems Engineering)
- [ ] 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: Differential equation
const ARTICLE = "Differential_equation";
let p1, p2, p3;
function setup() {
createCanvas(720, 520);
pixelDensity(2);
// physics-dynamics sim — create the three sliders described above:
// Rate k (-2–2), Initial value y0 (-10–10), Time step dt (0.01–0.5).
// TODO: build labelled sliders in the control region (below the drawing region).
}
function draw() {
background(248);
// TODO: read p1/p2/p3; render the physics-dynamics in the drawing region and respond live.
// TODO: HUD — title + en.wikitube.io/wiki/Differential_equation + control hints + live readouts.
}
```
---
Back to Systems Engineering
---
<!-- SEMIOTIC-PROFILE:START -->
## 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 (3):** 🟡 system (4) · 🟢 exponential (1) · 🟢 stability (1)
**Enter by sign:** Systems Semiotic Gateway · Alphabetum · Icon Registry · ← Systems Portal
<!-- SEMIOTIC-PROFILE:END -->
<!-- REAL-GENERATIVE-MEDIA:START -->
## Wikipedia : Wikitube
**Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Differential_equation) : [Wikitube](https://en.wikitube.io/wiki/Differential_equation)
## Previous hub tags
Tree parents: [[Control_theory]] · [[Decision_theory]] · [[Dynamical_system]] · [[Game_theory]] · [[Graph_theory]] · [[Information_theory]] · [[Operations_research]] · [[Phase_space]] · [[Systems_engineering]].
Legacy hubs: none.
---
*Sources: 1 legacy note. Minted wave 1, 2026-07-30 (v1.6 order).*