Question

State whether the given statement is True or False
In a circuit of Resistance $R$, self-inductance $L$,the current $i$ is given by $L\frac{di}{dt}+Ri=Ecospt$.where $E$ and $p$ are constants.The current $i$ at time $t$ is $i=\frac{E}{L^2{p}^2+R^2}+[Lpsinpt+Rcospt]+k{e}^{-Rt/L}$

• A. True
• B. False

Answer 1

The correct option is A True

Given differential equation is $\frac{di}{dt}+\frac{iR}{L}=E\cos pt$,

This is a Linear differential equation,

The integrating factor is,$e^{\frac{Rt}{L}}$,

The solution is $i{e}^{\frac{Rt}{L}}=\int e^{\frac{Rt}{L}}(E\cos pt)dt$,

Solving the integral gives,

We can solve it by Integration by parts where $E\cos pt=uand{e}^{-\frac{Rt}{L}}=v$

$i=\frac{E}{L^2{p}^2+R^2}+(Lp\sin pt+R\cos pt)+k{e}^{-\frac{Rt}{L}}$,

Where $k$ is the integration constant.