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 2020-11-20 Answered
Solve the general initial value problem modeling the LR circuit, dldt+RI=E,I(0)=I0, where E is a constant source of emf.
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Expert Answer

Corben Pittman
Answered 2020-11-21 Author has 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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New questions

Linear multivariate recurrences with constant coefficients
In the theory of univariate linear recurrences with constant coefficients, there is a general method of solving initial value problems based on characteristic polynomials. I would like to ask, if any similar method is known for multivariate linear recurrences with constant coefficients. E.g., if there is a general method for solving recurrences like this:
f ( n + 1 , m + 1 ) = 2 f ( n + 1 , m ) + 3 f ( n , m ) f ( n 1 , m ) , f ( n , 0 ) = 1 , f ( 0 , m ) = m + 2.
Moreover, is their any method for solving recurrences in several variables, when the recurrence goes only by one of the variables? E.g., recurrences like this:
f ( n + 1 , m ) = f ( n , 2 m ) + f ( n 1 , 0 ) , f ( 0 , m ) = m .
This second question is equivalent to the question, if there is a method of solving infinite systems of linear univariate recurrences with constant coefficients. That is, using these optics, the second recurrence becomes f m ( n + 1 ) = f 2 m ( n ) + f 0 ( n 1 ) , f m ( 0 ) = m , m = 0 , 1 , .
I am not interested in a solution of any specific recurrence, but in solving such recurrences in general, or at least in finding out some of the properties of possible solutions. For instance, for univariate linear recurrences, each solution has a form c 1 p 1 ( n ) z 1 n + + c k p k ( n ) z k n ,, where c i 's are constants, p i 's are polynomials and z i 's are complex numbers. Does any similar property hold for some class of recurrences similar to what I have written?
I have been googling a lot, but have found only methods for some very special cases (in monographs on partial difference equations, etc.), but nothing general enough. I am not asking for a detailed explanation of any method, but references to the literature would be helpful. I don't know much about transforms (like discrete Fourier transform or z-transform), but I found certain hints that there could be a method based on these techniques. Is it possible to develop something general enough using transform, i.e., is the study of transforms worth an effort (in the context of solving these types of recurrences)? However, it seems to me that the generalization of the characteristic polynomial method (perhaps, some operator-theoretic approach) could lead to more general results. Has there been any research on this topic?