Initial value problem

Definition

Let $\Omega$ be an open subset of $\mathbb{R} \times \mathbb{R}^m$, let $f : \Omega \to \mathbb{R}^m$ be a function, and let $(t_0, y_0) \in \Omega$. The initial value problem for $f$ at $(t_0, y_0)$ is the system $$y'(t) = f(t, y(t)), \qquad y(t_0) = y_0.$$ A solution of the problem is a solution $y : I \to \mathbb{R}^m$ of the equation $y' = f(t, y)$ on an open interval $I$ with $t_0 \in I$, such that $y(t_0) = y_0$. The condition $y(t_0) = y_0$ is the initial condition.

Remarks

For an equation of order $n$ in explicit form, the initial conditions are the values $y(t_0), y'(t_0), \dots, y^{(n-1)}(t_0)$. The reduction to a first-order system changes this problem into the problem above. The problem $y' = y$, $y(0) = 1$ has the unique solution $y(t) = e^t$. The Picard–Lindelöf theorem gives a unique local solution when $f$ is continuous and Lipschitz in $y$. The Peano existence theorem gives a solution when $f$ is only continuous.

Explanations

L1 human

A differential equation alone usually has many solutions. The initial condition says where the solution starts, and this picks one solution. Suppose that you know the speed of a car at each moment and its position at the start. Then you know its position at each moment.