Assume that the coils have zero winding resistance values, the mesh equations of the circuit shown in Fig. are given as

\({v_1}\left( t \right) = L\frac{{di_1}}{{dt}} - M\frac{{d{i_2}}}{{dt}}\)

\({v_2}\left( t \right) = \; M\frac{{d{i_1}}}{{dt}} - {L_2}\frac{{d{i_2}}}{{dt}}\)

\({v_1}\left( t \right) = - L\frac{{d{i_1}}}{{dt}} - M\frac{{d{i_2}}}{{dt}}\)

\({v_2}\left( t \right) = M\frac{{d{i_1}}}{{dt}} - {L_2}\frac{{d{i_2}}}{{dt}}\)

\({v_t}\left( t \right) = - {L_1}\frac{{d{i_2}}}{{dt}} - M\frac{{d{i_2}}}{{dt}}\)

\({v_2}\left( t \right) = M\frac{{d{i_1}}}{{dt}} + {L_2}\frac{{d{i_2}}}{{dt}}\)

\({v_t}\left( t \right) = {L_1}\frac{{d{i_1}}}{{dt}} + M\frac{{d{i_2}}}{{dt}}\)

\({v_2}\left( t \right) = - M\frac{{d{i_1}}}{{dt}} + {L_2}\frac{{d{i_2}}}{{dt}}\)