Solve the general initial value problem modeling the LR circuit, L frac{dI}{dt} + RI=e, I(0)=I_{0} where E is a constant source of emf.

waigaK 2021-02-02 Answered
Solve the general initial value problem modeling the LR circuit, L dIdt + RI=e, I(0)=I0 where E is a constant source of emf.
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Expert Answer

Jaylen Fountain
Answered 2021-02-03 Author has 170 answers

Step 1

Let our equation be LI(t) + RI(t)=E () where are I - moving charge L - inductor R - resistor E - const First, divide both sides of equation with L LI(t) + RI(t)= :L
I(t) + RL I(t)= EL We have first order linear differntial equation. To solve her, we must find integration factor μ(t). First, let's define function a(t) a(t)= RL We will get integration factor using next formula μ(t)=e a(t)dt=e RL dt=eRLt Now, multiply both sides of our equation with integration factor I(t) + RL I(t)= EL eRLt
eRLtI(t) + eRLt 1RCI(t)=eRLt El

Step 2

eRLtI(t) + eRLt RLI(t)=eRLt EL
= ddt (eRLtI(t))
ddt (eRLtI(t))=eRLt EL

Integrate both sides of equation  ddt (eRLtI(t))dt=  eRLt ELdt

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