# Completely-integrable differential equation

An equation of the form

$$\tag{* } \omega \equiv \ \sum _ {i = 1 } ^ { n } P _ {i} ( x) dx ^ {i} = 0,\ \ P _ {i} \in C ^ {1} ,$$

for which an $( n - 1 )$- dimensional integral manifold passes through each point of a certain domain in the space $\mathbf R ^ {n}$. A necessary and sufficient condition for complete integrability of the differential equation (*) is the Frobenius condition $\omega \wedge d \omega = 0$, where $\wedge$ is the symbol of the exterior product . If $n = 3$, this condition has the form:

$$P _ {1} \left ( \frac{\partial P _ {3} }{\partial x ^ {2} } - \frac{\partial P _ {2} }{\partial x ^ {3} } \right ) + P _ {2} \left ( \frac{\partial P _ {1} }{\partial x ^ {3} } - \frac{\partial P _ {3} }{\partial x ^ {1} } \right ) + P _ {3} \left ( \frac{\partial P _ {2} }{\partial x ^ {1} } - \frac{\partial P _ {1} }{\partial x ^ {2} } \right ) =$$

$$= 0.$$

Instead of equation (*) the following system of equations is sometimes considered :

$$dx ^ {i} = \ \sum _ {j = 1 } ^ { {n } - 1 } a _ {j} ^ {i} ( x, t) dt ^ {j} ,\ \ i = 1 \dots n.$$

In this case the conditions of complete integrability assume the form:

$$\sum _ {l = 1 } ^ { n } \frac{\partial a _ {j} ^ {i} }{\partial x ^ {l} } ( x, t) a _ {k} ^ {l} ( x, t) + \frac{\partial a _ {j} ^ {i} }{\partial t ^ {k} } ( x, t) =$$

$$= \ \sum _ {l = 1 } ^ { n } \frac{\partial a _ {k} ^ {i} }{\partial x ^ {l} } ( x, t) a _ {j} ^ {l} ( x, t) + \frac{\partial a _ {k} ^ {i} }{\partial t ^ {j} } ( x, t),$$

$$i = 1 \dots n ; \ j , k = 1 \dots n .$$

The family of integral manifolds of a completely-integrable differential equation is a foliation .

How to Cite This Entry:
Completely-integrable differential equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Completely-integrable_differential_equation&oldid=46424
This article was adapted from an original article by L.E. Reizin' (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article