# Differential Equations and Circuits

1. May 4, 2012

### nikki__10234

So, I'm learning how to solve LR, RC, LC etc. types of circuits using differential equations. I understand how to do the math with differential equations, but I am confused as to why the variables are split in the way they are.

For example, for an LR circuit you have the equation
L(dI/dt)+RI=V

and then the book integrates both sides:
∫(dI/(V-IR))=∫(dt/L)
and so on....

It is justified to group the I term with the dI, but I don't understand why the L, V, and R terms are placed as they are? Could you get the same result if they were not arranged in this way since the integral is not being taken in terms of V, R or L?

2. May 4, 2012

### tiny-tim

welcome to pf!

hi nikki! welcome to pf!
the V has to stay with the IR …

can you see any way of getting the IR over onto the LHS (with the dI), without the V coming with it?​

but the L could go either side

3. May 5, 2012

### haruspex

The only requirement is that any variables which actually vary in the range of integration stay inside an integral, but it will help if it varies as a function of the variable of integration with which it is placed.
E.g. if L = L(t) then it would best be placed in the integral .dt as above, whereas if R is a constant then you could just as easily write
∫(dI/(V/R-I))=R∫(dt/L)
OTOH, if V = V(t) it's going to get tricky ;-).