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

Cabiolab

Cabiolab

Answered question

2020-11-20

Solve the general initial value problem modeling the LR circuit, dldt+RI=E,I(0)=I0, where E is a constant source of emf.

Answer & Explanation

Corben Pittman

Corben Pittman

Skilled2020-11-21Added 83 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)=E:L
I(t)+RLI(t)=EL

We have first order linear differential 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)=ea(t)dt=eRLdt=eRLt

Now, multiply both sides of our equation with integration factor

I(t)+RLI(t)=ELeRLt
eRLtI(t)+eRLt1RCI(t)=eRLtEL

Step 2

eRLtI(t)+eRLtRLI(t)=eRLtRL
=ddt(eRLtI(t))
ddt(eRLtI(t))=eRLtEL

Integrate both sides of equation

ddt(eRLtI(t))dt=eRLtELdt
eRLtI(t)=ELeR

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