mathematics//differential equations//initial condition

An initial condition is the starting state that picks one solution out of the family a differential equation has. The equation has one solution for each initial condition: it fixes the rule, and the starting point fixes which member of the family the system follows.

An initial condition is the starting state that picks one solution out of the family a differential equation has. The equation has one solution for each initial condition: it fixes the rule, and the starting point fixes which member of the family the system follows.

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).

equationẋ = 0.50 x startx(0) = 1.00 One equation, a family of solutions: every curve follows the slopes of the field, and the starting point picks which curve is the solution.

Drag the green dot anywhere: the curve through it is the solution, every faint curve is the same formula with another CCC, and a negative kkk turns growth into decay.

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, the family a phase portrait draws; what the system does from a given point needs the initial condition.

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).

An uncertain initial condition is one of the homes of model error, and the dynamics decide what happens to it. A stable system forgets it, at the pace of its slowest mode; an unstable one amplifies it exponentially, and a chaotic one amplifies it while staying bounded, so that a thousand times better measurement of the start buys only a few more seconds of forecast (chaos). That is why a filter starts from an initial estimate with an honest uncertainty attached (initial covariance) and lets the measurements wash it out.

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.