Ordinary differential equations Concept
Differential equation
Summary
A differential equation is an equation that relates an unknown function to one or more of its derivatives. A solution is a function that makes the equation true at each point of its domain. If the unknown function depends on one variable, the equation is an ordinary differential equation. If it depends on several variables and the equation has partial derivatives, it is a partial differential equation.
Remarks
The order of a differential equation is the highest order of a derivative in it. The equation is a differential equation of order . Its solutions are the functions with a constant . Differential equations describe how a quantity changes. Many laws of physics, biology and economics have this form.
Where this leads
169 pages build on this one.
- Picard–Lindelöf theorem in integral form
- Non-uniqueness for a non-Lipschitz equation
- Exponential growth and decay
- Picard operator
- Lotka–Volterra equations
- Integral equation form of an initial value problem
- Maximal solution
- Periodic orbit
- Integrating factor
- Uniqueness of solutions for Lipschitz vector fields
- Flow of a vector field
- Newton's law of cooling
- Reduction of order
- C^1 vector fields satisfy the Picard–Lindelöf hypotheses
- Lyapunov function
- Solution of an ordinary differential equation
- Phase plane
- Asymptotic stability
- LaSalle's invariance principle
- Solution formula for linear first-order equations
- Initial value problem
- Autonomous system
- Phase line
- Power series solution
- Explicit and implicit differential equations
- An iterate of the Picard operator is a contraction
- Eigenvalue method for linear systems
- Lyapunov stability
- Peano existence theorem
- Order of a differential equation
- Liouville's formula
- Linearisation at an equilibrium
- Linear equation with constant coefficients
- System of linear differential equations
- Solutions for complex conjugate roots
- Global existence for globally Lipschitz vector fields
- Trajectory
- Hamiltonian system
- Hyperbolic equilibrium point
- Logistic equation
- Integrating factor for a non-exact equation
- Singular solution
- Riccati equation
- Blow-up in finite time
- Exact equation
- Limit cycle
- Homogeneous linear equation
- Poincaré–Bendixson theorem
- Superposition principle
- Continuous dependence on initial data
- Equilibrium point
- General solution of an inhomogeneous linear equation
- Solution space of a homogeneous linear equation has dimension n
- Fundamental system of solutions
- Solutions for distinct real roots
- Solutions for repeated roots
- Lyapunov's stability theorem
- Picard–Lindelöf hypotheses
- Euler method
- Underdamped
- Solving initial value problems with the Laplace transform
- Homogeneous first-order equation
- Wronskian test for linear independence of solutions
- Fundamental matrix
- Test for exactness
- Classification of equilibria of planar linear systems
- Picard–Lindelöf theorem
- Ordinary differential equation
- Linear differential equation of order n
- Characteristic equation
- Picard iteration
- Slope field
- Damped harmonic oscillator
- Variation of constants formula for linear systems
- Hartman–Grobman theorem
- Pendulum equation
- Stability of linear systems via eigenvalues
- General solution and particular solution
- First integral
- Phase portrait
- Clairaut's equation
- Reduction of a higher-order equation to a first-order system
- Separable equation
- Abel's identity
- Homogeneous linear system with constant coefficients
- Time translation of solutions of an autonomous system
- Cauchy–Euler equation
- Nullcline
- Forced oscillation and resonance
- Method of undetermined coefficients
- Variation of parameters
- Matrix exponential as the solution of a linear system
- Harmonic oscillator
- Linear first-order equation
- Bernoulli equation
- Liénard's theorem
- Hopf bifurcation
- Transcritical bifurcation
- Osgood uniqueness criterion
- von Neumann stability analysis
- Feigenbaum–Cvitanović functional equation
- Bendixson–Dulac theorem
- Carathéodory's existence theorem
- Center manifold theorem
- Limit set
- Spectral decomposition theorem
- Frobenius method
- Method of lines
- Comparison theorem
- Invariant set
- Reduction of an initial value problem to a Volterra equation
- Alternating-direction implicit method
- Crisis
- Linear multistep method
- Stiff equation
- Melnikov method
- Euler's elastica
- Homoclinic orbit
- Euler–Maruyama method
- Liénard equation
- Lax equivalence theorem
- Anosov flow
- Heteroclinic orbit
- Feigenbaum constants
- Center manifold
- Intermittency
- Differential-algebraic system of equations
- Floquet theory
- Bogdanov–Takens bifurcation
- Bifurcation diagram
- Abel equation of the first kind
- Runge–Kutta methods
- Poincaré map
- Differentiable dependence on parameters
- Routh–Hurwitz stability criterion
- Structural stability
- Crank–Nicolson method
- Homoclinic bifurcation
- Rössler attractor
- Hyperbolic set
- Symplectic integrator
- Kneser's theorem on solution sets
- Separation of variables
- Bifurcation theory
- Partial differential equation
- Duhamel's principle
- Annihilator method
- Logistic map
- Mild solution
- Heun's method
- Phase space
- Axiom A
- Duffing equation
- Saddle-node bifurcation
- Magnus expansion
- Pitchfork bifurcation
- Exponential stability
- Smale's fourteenth problem
- Jakobson's theorem
- d'Alembert's equation
- Van der Pol oscillator
- Stable manifold theorem
- Lorenz system
- Lie symmetry method
- Anosov diffeomorphism
- Stability conjecture
- Singer's theorem
- Hill differential equation
- Normal form
Further reading
- Web https://en.wikipedia.org/wiki/Differential_equation, lead section
About this page
| Tag | 003R0V5M7V |
|---|---|
| URI | m2://003R0V5M7V |
| Kind | concept |
| Writing | W1 stated |
| MSC | 34A99 |
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Special cases
- Ordinary differential equation
- Stochastic differential equation
- Partial differential equation
- Fractional differential equation