Solving a Series RLC Transient Analysis Problem

[Junior]

Let's say we have a free-source series RLC circuit. The differential equation that describes the behavior of the transient is given by $$Ri+L\frac{di}{dt}+V=0$$, where V is the potential difference across the capacitor. But I have trouble understanding the way this equation is developed. For example, suppose that we have the following situation:

But if I apply the loop rule, I get:

$$-V+Ri+L\frac{di}{dt}=0$$

What is wrong here? In the book they apply the loop rule to the same circuit above, but with the polarity of the capacitor inverted. But doesn't the current leave the positive terminal of it?

Thanks in advance!

 

[andrewkirk]

Let's say we have a free-source series RLC circuit.

I presume you mean a source free circuit.

Re the equation, they don't say in which direction they are measuring the PD across the capacitor. Perhaps they are measuring it as (potential of lower plate) minus (potential of upper plate), in which case the given equation would work.