I'm trying to solve $Y\left( x\right) A\left( x\right) +Y^{\prime }\left( x\right) =B\left( x\right) $ in the Banach algebra of bounded operators. Multiplying both sides by $\exp \left( \int^{x}A\left( t\right) dt\right) $ doesn't have the same consequence as in commutative Banach algebras. So, what can we do to solve this differential equation ?
Thank you !