# Mesh Current Method Circuit Analysis For AC Transient and Steady State Solution of RL Circuit

Hi everyone, I'm trying to find the currents through each branch for the above circuit using mesh current method (as I don't really know how else to do it), and end up with two simultaneous differential equations:

$$Mesh 1: R_si_1+R_1(i_1-i_2)+L_1\frac{d}{dt}(i_1-i_2)=155cos(377t)$$ $$Mesh 2: R_2i_2+L_2\frac{d}{dt}i_2+L_1\frac{d}{dt}(i_2-i_1)+R_1(i_2-i_1)=0$$

These simplify to:

$$Mesh 1: (R_s+R_1)i_1-R_1i_2+L_1\frac{d}{dt}i_1-L_1\frac{d}{dt}i_2=155cos(377t)$$ $$Mesh 2: -R_1i_1+(R_1+R_2)i_2-L_1\frac{d}{dt}i_1+(L_1+L_2)\frac{d}{dt}i_2=0$$

The problem I am having is that I cannot figure out a linear combination of these equations to eliminate either i_1 or i_2 variables to solve the system.

Am I going about this the wrong way? Or is there an easier way to solve this?

I can readily find the steady state solution reducing everything into impedances but I need to find the transient solution as well.