Ordinary Differential Equations

An equation that contains an unknown function of one variable and one or more of its derivatives.

Contrast with Partial Differential Equation

Eg, let $y=f(x)$

Then $y^{″}+y^{\prime }+2y=2x$ is a second order differential equation

Alternatively,

where $u$, $u^{\prime }$, and $u^{″}$ are position, velocity, acceleration at time $t$, $m$ is the mass, $F$ is force acting on object.

In this equation, Independent Variable is time $t$, and Dependent Variable is $u$.

Finding Independent and Dependent Variables

See also Order of a Differential Equation

ODEs can be written in Implicit Form

Solution to an ODE is a function that satisfies $y$, will tend to have a constant
Solving Ordinary Differential Equations
Solving Ordinary Differential Equations Numerically

Initial Value Problem

Autonomous ODE
Non-Autonomous ODE

nth Order Linear ODE

Linear vs Non-linear

ODE Exact Equation Theorem

Solving Second Order Linear Constant Coefficient Homogenous ODEs

See Also:
Separable First Order ODEs
Applications of Separable ODEs