Answer to Question #280615 in Differential Equations for Feline

Question

An RL circuit has an emf of 20 volts, a resistance of 20 ohms, an inductance of 5 H with an initial current of 4 mA. What is the current after 0.05 seconds?

Answer 1

Differential equation of a LR circuit 

$Li'+Ri=U$

Given $U=20V, R=20\ Ohm, L=5\ H$

Substitute

$5i'+20i=20$ $i'+4i=4$

Integrating factor

$\mu(t)=e^{\int4dt}=e^{4t}$

$e^{4t}(i'+4i)=4e^{4t}$

$d(e^{4t}i)=4e^{4t}dt$

Integrate

$\int d(e^{4t}i) =\int 4e^{4t}dt$

$e^{4t}i=e^{4t}+C$

$i(t)=1+Ce^{-4t}$

$i(0)=4mA=4\times 10^{-3} A$

$4\times 10^{-3} =1+C$

$C=-0.996$

$i(t)=1-0.996e^{-4t}$

$i(0.05)=1-0.996e^{-4(0.05)}$

$i(0.05)=184.5\ mA$