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Circuit Under Study

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.

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It would be easier to convert both the differential equations to laplace domain and then solve for i1(s) and i2(s). Take inverse laplace then to find i1(t) and i2(t).

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