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  "body": "\n## Definition\n\nLet $\\Omega$ be an open subset of $\\mathbb{R} \\times \\mathbb{R}^m$, let $f : \\Omega \\to \\mathbb{R}^m$ be a function, and let $(t_0, y_0) \\in \\Omega$.\nThe initial value problem for $f$ at $(t_0, y_0)$ is the system\n$$y'(t) = f(t, y(t)), \\qquad y(t_0) = y_0.$$\nA solution of the problem is a [[solution|04SW7WECS8]] $y : I \\to \\mathbb{R}^m$ of the equation $y' = f(t, y)$ on an open interval $I$ with $t_0 \\in I$, such that $y(t_0) = y_0$.\nThe condition $y(t_0) = y_0$ is the initial condition.\n\n## Remarks\n\nFor 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|0TH9CPW3VS]] changes this problem into the problem above.\nThe problem $y' = y$, $y(0) = 1$ has the unique solution $y(t) = e^t$.\nThe [[Picard–Lindelöf theorem|0P6KRD5B7A]] gives a unique local solution when $f$ is continuous and Lipschitz in $y$. The [[Peano existence theorem|098YP831QY]] gives a solution when $f$ is only continuous.\n\n## Explanations\n\n### L1 human\n\nA differential equation alone usually has many solutions.\nThe initial condition says where the solution starts, and this picks one solution.\nSuppose 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.\n",
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