# 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 $$u'' = A_iu$$.

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