Complex 2R, L, C, voltage circuit

Given that the voltage source is defined by ${\displaystyle V_{s}(t)=cos(t)}$ and the current source is defined by ${\displaystyle I_{s}(t)=2sin(t-{\frac {\pi }{3}})}$, find all other voltages, currents and check power.

Circuit marked up for analysis

Knowns, Unknowns and Equations

Knowns: ${\displaystyle V_{s},R_{1},R_{2},L,C,I_{s}}$

Unknowns: ${\displaystyle v_{1},v_{2},v_{C},v_{L},i_{1},i_{2},i_{C},i_{L}}$

Equations:

${\displaystyle v_{1}=R_{1}*i_{1}}$

${\displaystyle v_{2}=R_{2}*i_{2}}$

${\displaystyle i_{C}=C*{d \over dt}v_{C}}$

${\displaystyle v_{L}=L*{d \over dt}i_{L}}$

${\displaystyle i_{1}+I_{s}-i_{2}-i_{C}=0}$

${\displaystyle i_{c}-I_{s}-i_{L}=0}$

${\displaystyle v_{1}+v_{2}-V_{s}=0}$

${\displaystyle v_{C}+v_{L}-v_{2}=0}$

Solutions and Analysis

numeric solution calculated in the phasor domain

attempt at a Laplace solution

simulation at circuitlab