ClearAll["Global`*"]
;
(* Do as a regular differential equation *)
R=30;L=4.0;ω=10;V0 = 1000.0;ϕ=0;
v2[t_]=V0 Cos[10 t];
ans=DSolve[{v2[t]-v1[t]== R i1[t], v1[t]== L i1'[t],i1[0]==((R L )/ω)},{i1[t],v1[t]},t];
(* ... I picked an initial condition that was easy ... *)
i1[t_]=i1[t]/.ans[[1]]//FullSimplify//TrigFactor;
Print["From Diff EQ,           i = ",i1[t]]
θ = ArcTan[((ω L)/R)];  (* copied from in-class stuff about complex numbers *)
i2[t_] = (V0/(Sqrt[(R^2)+(ω^2)(L^2)]))Cos[ω t -θ];
Print["From 'complex' class,   i = ",(V0/(Sqrt[(R^2)+(ω^2)(L^2)]))," Cos[",ω," t - ",θ,"]"]
Print["Which MMA rearranges to   i = ",i2[t]]
Print["We found    θ = ",θ]
(* Now, Complex notation in Mathematica *)
R=30;L=4;ω=10;V0 = 1000;ϕ=0;
V3=V0 Cos[ω t]+V0  I Sin[ω t];
Ztot = R + I ω L;
i3[t_]=i3/.Solve[V3== i3 Ztot,i3][[1]]//ComplexExpand;
Print["Next, using this input:  V = ",V3//TrigToExp];
Print["and solving with Euler Notation, MMA says i = ",i3[t]//TrigToExp]
Print["or, i3 = Re[i] = ",Re[i3[t]]//ComplexExpand]
Print["From plot, all three currents are the same..."]
Print["but remember... only amplitude and phase matter here!"]
Plot[{i1[t],i2[t],Re[i3[t]]},{t,0,4}]