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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.

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    Yes, you should be using partial fraction decomposition and then reverse the Laplace transform. You might as well use $A=R/L$ and $B=1/(LC)$ if it helps you in the intermediate steps.2012-02-07

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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.

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I think the problem you are having is that none of the elements are suitably defined for your problem. Perhaps you need to treat the denominator as a quadratic function with three possible solutions: two distinct real roots, two repeating roots, and complex conjugates. Give that a try.

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    @Jon would you mind looking at a problem for me? Its over here: http://math.stackexchange.com/questions/1141213/solve-an-ordinary-sec0nd-order-differential-equation-for-an-lc-circuit-using-lap2015-02-10