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Functional magnetic resonance imaging (fMRI) conventionally relies on hemodynamics response induced some seconds after neural activation. Therefore, the temporal limitations of fMRI is the main challenge and we focused on the spin-lock sequence, which identifies neural magnetic field using magnetic resonance between the spin-lock pulse and the measurement target. This potentially addresses limitations of the traditional fMRI such as low temporal resolution and high static magnetic field. To enhance spin-lock sequences, it is essential to analyze magnetization dynamics during the spin-lock pulse. This study improved our previous proposal by presenting the particular solution of the Bloch equation, which allows us to consider the effect of $T_{1}$ relaxation. We also confirmed the novel solution with the comparison of numerical simulations without any approximation and discovered that the solution improved accuracy for $M_{y}$ and $M_{z}$ . The offset errors for $M_{y}$ and $M_{z}$ became smaller than $1\,\, {} \times {}10^{-5}$ times to the thermal equilibrium magnetization and the magnitude of errors was reduced to $3.2975\,\, {} \times {}10^{-3}$ and $1.3072\,\, {} \times {}10^{-1}$ times compared to our previous proposal. This improvement contributes to acquisition-integrated spin-lock sequences. There are still drawbacks due to the rotating wave approximation in our new proposal, but, it is applicable when the frequency bandwidth of spin-lock resonance is narrow, or the measurement target, such as a steady-state induced response, can be approximated by a single-frequency sine wave.