> [!NOTE] Definition (IVP for ODE) > An initial value problem is a [[Scalar Ordinary Differential Equation|scalar ODE]] of the form $x^{(n)}=f(t,x(t),x'(t),\dots,x^{(n-1)}(t))$for some function $(\alpha,\beta)\times \mathbb{R}^{k}\to \mathbb{R},$ together with a point $(t_{0},y_{0},y_{1},\dots,y_{n-1}) \in (\alpha,\beta)\times \mathbb{R}^{k}$known as the initial condition. # Properties **Solution**: A solution $\phi$ to an initial value problem is a [[Solution to Scalar Ordinary Differential Equation|solution of the ordinary differential equation]] that satisfies the initial condition, that is: $\phi(t_{0})=y_{0},\phi'(t_{0})=y_{1},\dots,\phi^{(n-1)}(t_{0})=y_{n-1}.$ **Existence and uniqueness of solution to IVPs:** There are several theorems that establish existence and uniqueness of solutions to initial value problems involving ODEs both locally and globally. The [[Picard–Lindelöf theorem|Picard–Lindelöf theorem]] guarantees a unique solution to $x'(t)=f(t,x(t))$ with $x(t_{0})=x_{0}$ on some interval containing $t_{0}$ if $f$ is continuous on a region containing $t_{0}$ and $y_{0}$ and satisfies the L on the variable $x.$ In its basic the theorem only guarantees a local result, though the latter can be extended to give a global result, for example, if the conditions of [[Uniqueness Theorem for Explicit First Order Initial Value Problem]] are met. An [[Example of Initial Value Problem with Infinitely Many Solutions|example of an initial-value problem with a non-unique solution]] is: $x'(t)=\sqrt{ x(t) },\quad x(0)=0.$