I'm trying to solve this second order differential equation using Laplace Transform. The Laplace transform of the equation is as follows:
$$I(s) = \frac{E}{s^2+ \frac{R}{L}s + \frac{1}{LC}}$$
I'm having trouble trying to bring it back to the time domain. Should I be using partial fractions with quadratic factors or is there a easier method to go abut this? Any help would be much appreciated.
Best Answer
Looking at the table here you will recognize three different possible behaviors. Let us see why. Consider the denominator. This is can be rewritten as
$$s^2+\frac{R}{L}s+\frac{1}{LC}=(s+\alpha)^2+\beta^2$$
where
$$\alpha=\frac{R}{2L} \qquad \beta=\sqrt{\frac{1}{LC}-\frac{R}{2L}}.$$
So, when $\frac{1}{LC}>\frac{R}{2L}$ you will recognize an exponentially decaying sine wave. When $\frac{1}{LC}=\frac{R}{2L}$ you will get just an exponential decay. When $\frac{1}{LC}<\frac{R}{2L}$ you will get an exponential decay multiplied by a hyperbolic cosine. All this can be deduced from the table I linked at the beginning of this answer.