In mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with specified value, called the initial condition, of the unknown function at a given point in the domain of the solution. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and A differential equation is a mathematical Equation for an unknown function of one or several variables that relates the values of the In Mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one Independent variable, and one or more of its In physics or other sciences, modeling a system frequently amounts to solving an initial value problem; in this context, the differential equation is an evolution equation specifying how, given initial conditions, the system will evolve with time. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Time evolution is the change of state brought about by the passage of Time, applicable to systems with internal state (also called stateful systems)

## Definition

An initial value problem is a differential equation

$y'(t) = f(t, y(t)) \quad\text{with}\quad f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$

together with a point in the domain of f

$(t_0, y_0) \in \mathbb{R} \times \mathbb{R},$

called the initial condition.

A solution to an initial value problem is a function y that is a solution to the differential equation and satisfies

y(t0) = y0.

This statement subsumes problems of higher order, by interpreting y as a vector. For derivatives of second or higher order, new variables (elements of the vector y) are introduced. In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change

More generally, the unknown function y can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions. In Mathematics, Banach spaces (ˈbanax named after Polish Mathematician Stefan Banach) are one of the central objects of study in Functional analysis In Mathematical analysis, distributions (also known as generalized functions) are objects which generalize functions and Probability distributions

## Existence and uniqueness of solutions

For a large class of initial value problems, the existence and uniqueness of a solution can be demonstrated.

The Picard-Lindelöf theorem guarantees a unique solution on some interval containing t0 if f and its partial derivative $\partial f/\partial y$ are continuous on a region containing t0 and y0. In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant The proof of this theorem proceeds by reformulating the problem as an equivalent integral equation. In Mathematics, an integral equation is an equation in which an unknown function appears under an Integral sign The integral can be considered an operator which maps one function into another, such that the solution is a fixed point of the operator. The Banach fixed point theorem is then invoked to show that there exists a unique fixed point, which is the solution of the initial value problem. The Banach Fixed point theorem (also known as the contraction mapping theorem or contraction mapping principle) is an important tool in the theory of

An older proof of the Picard-Lindelöf theorem constructs a sequence of functions which converge to the solution of the integral equation, and thus, the solution of the initial value problem. Such a construction is sometimes called "Picard's method" or "the method of successive approximations". This version is essentially a special case of the Banach fixed point theorem.

Hiroshi Okamura obtained a necessary and sufficient condition for the solution of an initial value problem to be unique. Hiroshi Okamura ( Japanese: 岡村 博 Okamura Hiroshi; November 10, 1905 – September 3, 1948) was a Japanese This condition has to do with the existence of a Lyapunov function for the system. In Mathematics, Lyapunov functions are functions which can be used to prove the stability of a certain fixed point in a Dynamical system or Autonomous

In some situations, the function f is not of class C1, or even Lipschitz, so the usual result guaranteeing the local existence of a unique solution does not apply. In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Mathematics, more specifically in Real analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a smoothness condition for functions The Peano existence theorem however proves that even for f merely continuous, solutions are guaranteed to exist locally in time; the problem is that there is no guarantee of uniqueness. In Mathematics, specifically in the study of Ordinary differential equations the Peano existence theorem, Peano theorem or Cauchy-Peano theorem The result may be found in Coddington & Levinson (1955, Theorem 1. 3) or Robinson (2001, Theorem 2. 6).

## Example

The general solution of

$y'+3y=6t+5,\qquad y(0)=3$

can be found to be

y(t) = 2e − 3t + 2t + 1.

Indeed,

 y' + 3y = (d / dt)(2e − 3t + 2t + 1) + 3(2e − 3t + 2t + 1) = ( − 6e − 3t + 2) + (6e − 3t + 6t + 3) = 6t + 5.