$ L\frac{di}{dt}+Ri=E\cos pt$

Dividing throughout by L,we get

$\frac{di}{dt}+\frac{R}{L}i=\frac{E}{L\cos pt} $

This is a linear differential equation in i with $P=\frac{R}{L}$ and $Q=\frac{E}{L} cospt$

$ I.F=e^{\int\frac{R}{L}dt}=e^{\frac{Rt}{L}}$

The solution is

$ie^{\frac{Rt}{L}}=\int e^{\frac{Rt}{L}}\frac{E}{L}\cos ptdt$

$ ie^{\frac{Rt}{L}}=\frac{E}{L}.\frac{e^{\frac{Rt}{L}}}{\left(\frac{R}{L}\right)^2+p^2}\left( \frac{R}{L}\cos pt+p\sin pt\right)+c$

$ ie^{\frac{Rt}{L}}=E.\frac{e^{\frac{Rt}{L}}}{R^2+p^2L^2}\left( R\cos pt+Lp\sin pt\right)+c$