mathematics//differential equations//ODE

An **ordinary differential equation** has one independent variable, almost always time, and unknowns that are functions of it. Everything that varies does so along one axis, and the derivatives are total derivatives.

An ordinary differential equation has one independent variable, almost always time, and unknowns that are functions of it. Everything that varies does so along one axis, and the derivatives are total derivatives.

The general form used across these notes is a system of first-order equations, x˙=f(x,u,t)\dot x=f(x,u,t)x˙=f(x,u,t), with xxx a vector. A higher-order equation is the same object after renaming derivatives as states: a second-order equation becomes two first-order ones by introducing the velocity, as the oscillator shows.

Whether ttt appears in fff splits the family into autonomous systems, whose field is fixed, and non-autonomous ones, whose field moves as time passes.

Its solutions are trajectories through a state space, one per initial condition; the geometry of all of them is the phase portrait. Where the unknown depends on space as well, the equation is a PDE.