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What is an expression for the transform current through resistor R ?(see attachment)

Mathematics
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1 Attachment
Is the source voltage an arbitrary function of time?
The current flows from the source through the first inductor, and then it branches off; part of the current flows through a purely capacitive branch. and the rest of it flows into the other inductor, which is in series with a resistor. At any instant, the voltages across the branches are the same. We can use this information and Kirchoff's laws to set up a system of equations to determine the currents throughout the circuit, including the current through the resistor.

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Other answers:

I will be doing some pencil-and-paper calculations on this, so it may be some time before my next post here.
Tank you for your time !!!!!!!
I get\[L^{2}Ci'''_R + RLCi''_R + Li'_R = e(t)\]for the time-domain equation. Since we are taking Laplace transforms of derivatives up to third-order, \[i''_R(0), i'_R(0),\]and\[i_R(0)\]enter into my calculations. Since nothing is stated about initial values for the current across the resistor, I keep those three numbers arbitrary, leaving them in symbolic form. When I take the Laplace transform of both sides of the equation I came up with, I get \[I_R(s) = E(s)/(L^{2}Cs^{3}+ RLCs^{2} + Ls)\] \[+ L^{2}Ci_R(0)s^{2}/(L^{2}Cs^{3} + RLCs^{2}+ Ls)\] \[+ (L^{2}Ci'_R(0) + RLCi_R(0))s/(L^{2}Cs^{3} + RLCs^{2} + Ls)\] \[+ (L^{2}Ci''_R(0) + RLCi'_R(0) + LI_R(0))/(L^{2}Cs^{3} + RLCs^{2} + Ls).\]I get\[L^{2}Ci'''_R + RLCi''_R + Li'_R = e(t)\]for the time-domain equation. Since we are taking Laplace transforms of derivatives up to third-order, \[i''_R(0), i'_R(0),\]and\[i_R(0)\]enter into my calculations. Since nothing is stated about initial values for the current across the resistor, I keep those three numbers arbitrary, leaving them in symbolic form. When I take the Laplace transform of both sides of the equation I came up with, I get \[I_R(s) = E(s)/(L^{2}Cs^{3}+ RLCs^{2} + Ls)\] \[+ L^{2}Ci_R(0)s^{2}/(L^{2}Cs^{3} + RLCs^{2}+ Ls)\] \[+ (L^{2}Ci'_R(0) + RLCi_R(0))s/(L^{2}Cs^{3} + RLCs^{2} + Ls)\] \[+ (L^{2}Ci''_R(0) + RLCi'_R(0) + LI_R(0))/(L^{2}Cs^{3} + RLCs^{2} + Ls).\]
Thank you !!!!!!!!
You're welcome :^)

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