Homework Help: Partial fractions!

1. Dec 2, 2009

mxpxer7

I'm sure this is a no brainer to someone, but here it is..

what is does the partial fraction of this look like in expanded form? Or how can I make it fit on the table of laplace transforms?

__(2s+1)__
(s-1)^2 + 1

Last edited: Dec 2, 2009
2. Dec 2, 2009

RedX

I think you'd get more replies (and better ones) if you posted this in the algebra forum, as this doesn't really have anything to do with differential equations. Your case would fall under the "irreducible quadratic factor in the denominator", as the denominator has zeroes s=1+i and s=1-i which aren't real:

So what you wrote is the simplest form if you want to use only real numbers.

3. Dec 2, 2009

mxpxer7

I apologize for putting this in the wrong section I'm using the answer for la place transforms so maybe this is where the calculus comes in, How can i write this so it fits into the table of laplace transforms?

4. Dec 2, 2009

RedX

$$F(s)=\frac{2s+1}{(s-1)^2+1}=2\frac{s-1}{(s-1)^2+1}+3\frac{1}{(s-1)^2+1}=G(s-1)$$

where $$G(s)=2\frac{s}{(s)^2+1}+3\frac{1}{(s)^2+1}$$

The inverse Laplace transform of G(s) is 2*cos(x)+3*sin(x). The inverse Laplace transform of G(s-1) is $$e^x$$[2*cos(x)+3*sin(x)] according to some of the properties of Laplace transforms and a shifted arguments.

5. Dec 3, 2009

trambolin

Nice shortcut...