The present study focuses the effects of Reynolds number Re and magnetic Reynolds number Rm on the transient behavior of the MHD flow. The incompressible, electrically conducting and viscous fluid flows through a long pipe subjected to magnetic field B0(t)=B0f(t). B0 is the intensity and f(t) is the time varying function of the magnetic field which is chosen as polynomial, trigonometric, exponential and logarithmic function to illustrate the problem parameters effects. The Re and Rm effects on the behavior of the flow at transient levels are studied with these functions by taking Hartmann number Ha value as 20. The unsteady MHD equations in coupled form are treated by using the dual reciprocity boundary element method (DRBEM). The study reveals that, when Re or Rm increases the time level where the flow elongates is postponed to a further time level. Moreover, the increase in Re flattens the flow as in the increase of Hartmann number. However, the increase in Rm increases the flow magnitude. The transient flow and induced current contours are demonstrated for several Re and Rm values. After the flow elongates, the flow and induced current lines preserve the behavior for polynomial, exponential and logarithmic type f(t) while trigonometric type f(t) causes the flow to show periodic behavior.