One of the limitations of the development of nuclear magnetic resonance (NMR) is the processing and interpretation of signals recorded by the modulation method. Existing descriptions of the magnetization dynamics in condensed media are usually approximate or purely numerical. In this work we present an analytical solution of the Bloch equations with a modulated field (H0 + Hmsin(ωmt)), in which the NMR signal is registered. The system is solved in matrix form using the Jordan decomposition of the exponential operator, and explicit expressions for the magnetization components are obtained in terms of incomplete gamma and hypergeometric functions. The obtained analytical solution allows analysing recorded signals and determining the ranges of registration parameters that ensure sufficient signal-to-noise ratio. The solution is validated by comparison with numerical modelling and experiment.