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In [[mathematics]], in the field of [[differential equation]]s, an '''initial value problem''' (also called '''the [[Cauchy problem]]''' by some authors) is an [[ordinary differential equation]] together with a specified value, called the '''initial condition''', of the unknown function at a given point in the domain of the solution. 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 [[time evolution|evolve with time]].

== Definition ==

An '''initial value problem''' is a differential equation
:<math>y'(t) = f(t, y(t)) \,</math> with <math> f: \Omega \subset \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n</math> where <math>\Omega \,</math> is an open set of <math>\mathbb{R} \times \mathbb{R}^n</math>,
together with a point in the domain of ƒ
:<math>(t_0, y_0) \in \Omega,</math>
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
:<math>y(t_0) = y_0. \,</math>

In higher dimensions, the differential equation is replaced with a family of equations <math>y_i'(t)=f_i(t, y_1(t), y_2(t), ...)</math>, and <math>y(t)</math> is viewed as the vector <math>(y_1(t), ..., y_n(t))</math>. More generally, the unknown function ''y'' can take values on infinite dimensional spaces, such as [[Banach space]]s or spaces of [[distribution (mathematics)|distribution]]s.

Initial value problems are extended to higher orders by treating the derivatives in the same way as an independent function, e.g. <math>y''(t)=f(t,y(t),y'(t))</math>.

== Existence and uniqueness of solutions ==

For a large class of initial value problems, the existence and uniqueness of a solution can be illustrated through the use of a calculator.

The [[Picard–Lindelöf theorem]] guarantees a unique solution on some interval containing ''t''<sub>0</sub> if ƒ is continuous on a region containing ''t''<sub>0</sub> and ''y''<sub>0</sub> and satisfies the [[Lipschitz continuity|Lipschitz condition]] on the variable ''y''.
The proof of this theorem proceeds by reformulating the problem as an equivalent [[integral equation]]. The integral can be considered an operator which maps one function into another, such that the solution is a [[Fixed point (mathematics)|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.

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. This condition has to do with the existence of a [[Lyapunov function]] for the system.

In some situations, the function ƒ is not of [[Smooth function|class ''C''<sup>1</sup>]], or even [[Lipschitz continuity|Lipschitz]], so the usual result guaranteeing the local existence of a unique solution does not apply. The [[Peano existence theorem]] however proves that even for ƒ merely continuous, solutions are guaranteed to exist locally in time; the problem is that there is no guarantee of uniqueness. The result may be found in Coddington & Levinson (1955, Theorem 1.3) or Robinson (2001, Theorem 2.6). An even more general result is the [[Carathéodory existence theorem]], which proves existence for some discontinuous functions ƒ.

==Examples==

A simple example is to solve <math>y' = 0.85 y</math> and <math>y(0) = 19</math>. We are trying to find a formula for <math>y(t)</math> that satisfies these two equations.

Start by noting that <math>y' = \frac{dy}{dt}</math>, so

: <math>\frac{dy}{dt} = 0.85 y</math>

Now rearrange the equation so that <math>y</math> is on the left and <math>t</math> on the right

: <math>\frac{dy}{y} = 0.85dt</math>

Now integrate both sides (this introduces an unknown constant <math>B</math>).

: <math>\ln | y | = 0.85t + B </math>

Eliminate the <math>\ln</math>

: <math> | y | = e^Be^{0.85t} </math>

Let <math>C</math> be a new unknown constant, <math>C = \pm e^B</math>, so

: <math> y = Ce^{0.85t} </math>

Now we need to find a value for <math>C</math>. Use <math>y(0) = 19</math> as given at the start and substitute 0 for <math>t</math> and 19 for <math>y</math>

: <math> 19 = C e^{0.85 * 0}</math>
: <math> C = 19 </math>

this gives the final solution of <math> y(t) = 19e^{0.85t}</math>.

;Second example
The solution of

: <math>y'+3y=6t+5,\qquad y(0)=3</math>

can be found to be

: <math>y(t)=2e^{-3t}+2t+1. \,</math>

Indeed,

: <math> \begin{align}
y'+3y &= \tfrac{d}{dt} (2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1) \\
&= (-6e^{-3t}+2)+(6e^{-3t}+6t+3) \\
&= 6t+5.
\end{align} </math>

==See also==
* [[Boundary value problem]]
* [[Constant of integration]]
* [[Integral curve]]

== References ==
<div class="references-small">
* {{cite book | author=Coddington, Earl A. and Levinson, Norman | title=Theory of ordinary differential equations | publisher=McGraw-Hill Book Company, Inc. | location=New York-Toronto-London | year=1955 }}
* {{cite book | author=[[Morris W. Hirsch|Hirsch, Morris W.]] and [[Stephen Smale|Smale, Stephen]] | title=Differential equations, dynamical systems, and linear algebra | publisher=Academic Press | location=New York-London | year=1974 }}
* {{cite journal | last=Okamura | first=Hirosi | title=Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano | journal=Mem. Coll. Sci. Univ. Kyoto Ser. A. | volume=24 | year=1942 | language=French | pages=21–28 }}
* {{cite book | author=Agarwal, Ravi P. and Lakshmikantham, V. | title=Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations | url=http://books.google.com/books?id=q4OkW4H8BCUC | series=Series in real analysis | volume=6 | year=1993 | publisher=World Scientific | isbn=978-981-02-1357-2}}
* {{cite book | author=Polyanin, Andrei D. and Zaitsev, Valentin F. | title=Handbook of exact solutions for ordinary differential equations | edition=2nd | publisher=Chapman & Hall/CRC | location=Boca Raton, FL | year=2003 | isbn=1-58488-297-2 }}
* {{cite book | last=Robinson | first=James C. | title=Infinite-dimensional dynamical systems: An introduction to dissipative parabolic PDEs and the theory of global attractors | publisher=Cambridge University Press | location=Cambridge | year=2001 | isbn=0-521-63204-8 }}
</div>

[[Category:Boundary conditions]]

[[cs:Počáteční podmínky]]
[[el:Αρχική τιμή]]
[[it:Problema ai valori iniziali]]
[[sv:Begynnelsevärdesproblem]]

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128.16.7.220: /* Problems and Limitations */ Bill replace * by multiply

en>Monkbot: /* The Framework of Manna and Waldinger */Fix CS1 deprecated date parameter errors

128.16.7.220: /* Problems and Limitations / Bill replace by multiply