Integrals of \(\nu\)-graded systems of ordinary differential equations
The problem of finding integrals of a system of n first-order differential equations is discussed: \[ Dy_ i=f_ i(y)\quad (i=1,...,n)\quad,\quad y=(y_ 1,...,y_ n) \] where the \(f_ i's\) are infinite series of rational functions in y and K, the field of coefficients, is a differential field. The author obtains an algorithm which yields all integrales of a system S which are in \(F_{\nu}\), if \(f_ i(y)\in F_{\nu}\) and \(\nu (f_ i(y))\geq \nu (y_ i)\) \((i=1,...,n)\), where \(\nu\) is some usual valuation of \(F=K(y)\) over K and \(F_{\nu}\) is the completion of F with respect to the given valuation. He shows how to use this algorithm to find rational or exponential integrals on the several examples. Also the author generalizes a result of Bruns for the N-body problem and of R. Liouville for the Euler equations.
integrals of system of first-order differential equations
scientific article; zbMATH DE number 4000065