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

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

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

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

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

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

$$C=-0.996$$

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

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

$$i(0.05)=184.5\ mA$$