I'm just starting a differential equations class and so far I am completely lost with these application problems. Specifically, in this one problem asks to find a differential equation for the current of a circuit. The problem:
Determine a differential equation for the current $i(t)$ if the resistance is $R$, the inductance is $L$, and the impressed voltage is $E(t)$.
The examples in the book mention using Kirchoff's second law, which the book defines as "the impressed voltage $E(t)$ on a closed loop must equal the sum of the voltage drops in the loop." But the book does not give an example of how to set up a DE using this law so I'm pretty lost. How do I even get started?
Best Answer
The voltage across a resistance $R$ with current $i(t)$ flowing through it is $i(t)R$ from Ohm's law. The voltage across an inductance $L$ with current $i(t)$ flowing through it is $L\frac{di(t)}{dt}$. If you had a capactitance $C$ with current $i(t)$, the voltage across it would be $\frac{1}{C}\int_{-\infty}^{t}i(t)dt$. I assume you have a circuit with a resistance $R$ in series with an inductor $L$ and an applied voltage $E(t)$. in that case we can write a single loop equation starting from the positive side of $E(t)$:
$$i(t)R + L\frac{di(t)}{dt} - E(t) = 0$$.