Rational solutions to algebraic differential equations. (English. Russian original) Zbl 0809.34008

Gorbuzov, V. N.; Nemets, V. S.

Differ. Equations 29, No. 10, 1454-1460 (1993); translation from Differ. Uravn. 29, No. 10, 1675-1683 (1993).

Let us consider the differential equation in the complex plane \[ \sum^ N_{i=0} B_ i(z) \prod^ n_{k=0} \{w^{(k)}\}^{\nu_{ki}}= 0,\tag{1} \] where \(B_ i(z)\) are polynomials of degree \(\deg B_ i(z)= b_ i\). The paper contains a list of conditions under which the equation (1) has (can have) rational solutions (2) \(w(z)= P(z)/Q(z)\), where \(P(z)= a_ p z^ p+\cdots+ a_ 0\), \(Q(z)= d_ q z^ q+\cdots+ d_ 0\). The characteristics of rational solutions (2) as the degree \((p,q)\), the difference \(p-q\) and the ratio \(a_ p/d_ q\) are expressed directly in terms of the parameters \(k\), \(\nu_{ki}\), \(b_ i\). The proofs are omitted.

Reviewer: A.Klíč (Praha)

MSC:

34M99

Ordinary differential equations in the complex domain

Keywords:

differential equation in the complex plane; rational solutions