mathematics//differential equations//initial condition

A differential equation does not have one solution. It has a family of them, one for each initial condition. The equation fixes the rule; where you start fixes which member of the family you get.

A differential equation does not have one solution. It has a family of them, one for each initial condition. The equation fixes the rule; where you start fixes which member of the family you get.

x˙=kx⟹x(t)=Cekt,C=x(0).\dot x=kx\quad\Longrightarrow\quad x(t)=Ce^{kt},\qquad C=x(0).x˙=kx⟹x(t)=Cekt,C=x(0).

Every value of CCC is a valid solution of the same equation. Only after fixing x(0)x(0)x(0) is the trajectory determined. The same holds for any system: x˙=f(x)\dot x=f(x)x˙=f(x) plus x(t0)=x0x(t_0)=x_0x(t0​)=x0​ is the initial value problem, and the equation without the condition is a statement about all trajectories at once.

This is exactly what a phase portrait shows. Each trajectory follows a very different path depending on where it begins, and drawing many of them on one picture is the whole point of the portrait: it is the family of solutions made visible.

Under mild regularity of fff the solution through a given point exists and is unique, which is why two trajectories of an autonomous system never cross: at the crossing they would share a state and have two different futures (autonomous system).

Higher order needs more data: a second-order equation needs position and velocity at t0t_0t0​, one condition per state variable of the equivalent first-order system (oscillator). For a PDE the "initial condition" is a whole profile, plus boundary conditions in space.