zbMATH — the first resource for mathematics

Integral representation of solutions for one singular equation containing the sum of commuting operators. (English. Russian original) Zbl 0829.34046
Differ. Equations 28, No. 5, 676-682 (1992); translation from Differ. Uravn. 28, No. 5, 831-838 (1992).
The abstract singular differential equation \(u'' + kt^{-1} u' = Au\), where \(k\) is a real number and \(A\) a linear (in general unbounded) operator in a Banach space is studied. Explicit solutions are constructed under the assumption that \(A = \sum^n_{i = 1} A_i\) with the operators \(A_i\) subject to the most general constraints that are necessary for solvability of the Cauchy problem for the equations \(u'' = A_iu\).

34G10 Linear differential equations in abstract spaces
34A05 Explicit solutions, first integrals of ordinary differential equations