Fig. $3$ shows an electrical network given in the time$-$domain.

$($a$)$ Write the nodal equations using $I_{s1},I_{s2},V_s,R_1,R_2,jL,$ and $(\frac{1}{j}C).$

$($c$)$ Using these equations, obtain the values of the node voltages $V_a,V_b$ and $V_c.$

$($d$)$ Hence obtain the values of the branch currents $I_1,I_2,I_3$ and $I_4.$

The various component values are given by: $V_S=1e^{j0}V,I_{s1}=1e^{j0}mA,I_{s2}=2e^{j0}mA,R_1=1k$ ohms, $R_2=1\mathrm{.}5k$ ohms, $j{x}_1=j1k$ ohms, $j{x}_2=-j1\mathrm{.}5k$ ohms.

$(8$ marks$)$