Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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.

share|improve this question
    
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. –  anon Feb 7 '12 at 10:54

1 Answer 1

up vote 2 down vote accepted

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.