Initial value problem

An initial value problem^[a] is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to solving an initial value problem. In that context, the differential initial value is an equation which specifies how the system evolves with time given the initial conditions.

Definition

An initial value problem is a differential equation

$y'(t)=f(t,y(t))$ with $f\colon \Omega \subset \mathbb {R} \times \mathbb {R} ^{n}\to \mathbb {R} ^{n}$ where $\Omega$ is an open set of $\mathbb {R} \times \mathbb {R} ^{n}$,

together with a point in the domain of $f$

$(t_{0},y_{0})\in \Omega$,

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

$y(t_{0})=y_{0}$.

In higher dimensions, the differential equation is replaced with a family of equations $y_{i}'(t)=f_{i}(t,y_{1}(t),y_{2}(t),\dotsc )$, and $y(t)$ is viewed as the vector $(y_{1}(t),\dotsc ,y_{n}(t))$, most commonly associated with the position in space. More generally, the unknown function $y$ can take values on infinite dimensional spaces, such as Banach spaces or spaces of distributions.

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

Existence and uniqueness of solutions

For a large class of initial asdwproblems, the existence and uniqueness dsa The Picard–Lindelöf theorem guarantees a unique solution on some interval containing t₀ if ƒ is continuous on a region containing t₀ and y₀ and satisfies the Lipschitsdwsz candidas] on the variable y.w an equivalent [as[integral equation. The integral can be considered an operator which maps one function into another, such that the solution is a [[Fixed point (mathematics)|fixsdes poasd An older proof of the Picard–Lindelöf theorem constructs a sequence of funcdtions 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" d Hiroshi Okamura obtaianed a [[necessary and sa

Examples

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

Start by noting that $y'={\frac {dy}{dt}}$, so

${\frac {dy}{dt}}=0.85y$

Now rearrange the equation so that $y$ is on the left and $t$ on the right

${\frac {dy}{y}}=0.85dt$

Now integrate both sides (this introduces an unknown constant $B$).

$\ln |y|=0.85t+B$

Eliminate the $\ln$

$|y|=e^{B}e^{0.85t}$

Let $C$ be a new unknown constant, $C=\pm e^{B}$, so

$y=Ce^{0.85t}$

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

$19=Ce^{0.85\cdot 0}$

$C=19$

this gives the final solution of $y(t)=19e^{0.85t}$.

Second example

The solution of

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

can be found to be

$y(t)=2e^{-3t}+2t+1.\,$

Indeed,

${\begin{aligned}y'+3y&={\tfrac {d}{dt}}(2e^{-3t}+2t+1)+3(2e^{-3t}+2t+1)\\&=(-6e^{-3t}+2)+(6e^{-3t}+6t+3)\\&=6t+5.\end{aligned}}$

Notes

[a] Also called a Cauchy problem by some authors

See also

• Boundary value problem
• Constant of integration
• Integral curve

References

• Coddington, Earl A.; Levinson, Norman (1955). Theory of ordinary differential equations. New York-Toronto-London: McGraw-Hill Book Company, Inc.
• Hirsch, Morris W. and Smale, Stephen (1974). Differential equations, dynamical systems, and linear algebra. New York-London: Academic Press.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Okamura, Hirosi (1942). "Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano". Mem. Coll. Sci. Univ. Kyoto Ser. A. (in French). 24: 21–28. MR 0031614.
• Agarwal, Ravi P.; Lakshmikantham, V. (1993). Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. Series in real analysis. Vol. 6. World Scientific. ISBN 978-981-02-1357-2.
• Polyanin, Andrei D.; Zaitsev, Valentin F. (2003). Handbook of exact solutions for ordinary differential equations (2nd ed.). Boca Raton, Florida: Chapman & Hall/CRC. ISBN 1-58488-297-2.
• Robinson, James C. (2001). Infinite-dimensional dynamical systems: An introduction to dissipative parabolic PDEs and the theory of global attractors. Cambridge: Cambridge University Press. ISBN 0-521-63204-8.