# Solving a first-order differential equation using Laplace Transform

everyone. I am trying to solve the following differential equation using the Laplace Transform:

$$L\frac{di}{dt} + Ri = E$$

I have managed to reduce it to the following equation:

$$I(s) = \frac{E}{sL(s+R/L)}$$

But I'm having a problem with it from there. Can somebody help me solve this please? Any help would be much appreciated.

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What you can do at this stage is use a decomposition in partial fractions to simplify your solution so that you can easily transform it back. –  Raskolnikov Feb 7 '12 at 9:17
Thanks Raskolnikov. –  D Brown Feb 7 '12 at 9:43

Your equation has a very simple solution given by

$$i(t)=Ae^{-\frac{R}{L}t}+\frac{E}{R}$$

being $A$ an arbitrary constant. Now, take a look at your Laplace transform

$$I(s)=\frac{E}{sL\left(s+\frac{R}{L}\right)}$$

and you will recognize two poles for $s=0$ and $s=-\frac{R}{L}$. The first pole will give rise to the constant term while the other one is just the exponential contribution. Indeed, here you can find a table of well-known transformed functions. This is essential in circuit analysis. Your case is the "exponential approach".

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Thanks a lot Jon! –  D Brown Feb 7 '12 at 9:43