mathematics//differential equations//ODE

An ODE (ordinary differential equation) is a differential equation with one independent variable, almost always time, and unknowns that are functions of it, and it is used to model a system whose state evolves along that one variable. Physics rarely says where something will be; it says how it changes, and an ODE is that statement written as a rule giving the state's rate of change from the state, the inputs and sometimes the time. The trajectory is what comes out of integrating the rule. The word *ordinary* says that derivatives are taken with respect to time only.

An ODE (ordinary differential equation) is a differential equation with one independent variable, almost always time, and unknowns that are functions of it, and it is used to model a system whose state evolves along that one variable. Physics rarely says where something will be; it says how it changes, and an ODE is that statement written as a rule giving the state's rate of change from the state, the inputs and sometimes the time. The trajectory is what comes out of integrating the rule. The word ordinary says that derivatives are taken with respect to time only.

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 tank filled from above and emptied through an orifice is one line of it, volume change equals inflow minus outflow (balance equation); the first-order system τx˙=−x+Ku\tau\dot x=-x+Kuτx˙=−x+Ku is one of the most used models in industry.

Reduction to first order turns any higher-order equation into that form by naming derivatives as states. A mass on a spring and damper, mp¨+cp˙+kp=Fm\ddot p+c\dot p+kp=Fmp¨​+cp˙​+kp=F, becomes p˙=v\dot p=vp˙​=v and v˙=(F−cv−kp)/m\dot v=(F-cv-kp)/mv˙=(F−cv−kp)/m once the velocity vvv is a variable, which is why the state of every mechanical system is position plus velocity (oscillator).

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.

Almost no interesting ODE is solved by hand. Solving the linear ones on paper is how one understands them; in practice they are stepped forward by a computer, and choosing the step and the method is its own subject (numerical integration). And when a system settles much faster than anything of interest, a static model (a table, a curve) is enough: the ODE pays when the transient matters.

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, which a computer chops back into a large ODE system.