We consider a broad class of linear operator equations that includes systems of ordinary differential equations, difference equations and fractional-order ordinary differential equations. This class also includes operator exponentials and powers, as well as eigenvalue problems and Fredholm integral equations. Many problems in engineering and the physical and natural sciences can be described by such operator equations. We generalise the fundamental matrix to a fundamental operator and provide a new explicit method for obtaining an exact series solution to these types of operator equations, together with sufficient conditions for convergence and error bounds. Illustrative examples are also given.