# Inverting a Laplace Transform w/non-factorable denominator

1. Oct 13, 2009

### monet A

1. The problem statement, all variables and given/known data

A 2nd order DE with a Delta Dirac forcing function, I have been asked to solve in my DE course.
y'' + 2y' + 10y = $$\delta$$(t-1)

Also given are IV's y'(0)=0, y(0)=0

2. Relevant equations

At this stage in my course I am being asked to only use the method of partial fractions to rewrite my Laplace equations into a recognisable invertable form I am not supposed to be using the integral definition to invert my solutions.

3. The attempt at a solution

The Transforms of each side I have so far obtained:

$$Y(s).(s^2 + 2s + 10) = e^{-x}$$

This gives me a solution for Y(s) -->
$$Y(s) = \frac{ e^{-x} }{s^2 + 2s + 10}$$

So I have $$Y(s) = e^{-x} . F(s)$$
$$e^{-x}$$ is no problem to invert but $$F(s)$$ has a quadratic in denominator which cannot be factored. I am at a loss as to how I can find any invertable form for it and for some reason I'm not aware of any other technique that is possible.

2. Oct 13, 2009

### rock.freak667

try completing the square for s2+2s+10

3. Oct 13, 2009

### monet A

Ok I feel stupid, I was thinking that completing the square would give me Y(s) terms on the RHS which I would have to divide out but I just tried it and now I don't know why I was thinking that.

I have factors but now I have a denominator with a dangling 6 that I don't know what to do with.

$$Y(s)s^2 + Y(s)2s + Y(s)4 = e^{-s} - Y(s)6$$
____

$$Y(s)((s+2)(s+2) + 6) = e^{-s}$$
So what now?

Last edited: Oct 14, 2009
4. Oct 14, 2009

### rock.freak667

While (s+2)2+6 will give you back the original function, that is not completing the square.

Remember you want s2+2s+10 =(s+1)2/+A, so what is A?

5. Oct 14, 2009

### monet A

Yeah, it's 9. Thanks for pointing out my mistake there, but also that's not the problem that I am having. THe thing is that the 9 has a factor of Y(s) so I have no choice but to return to the original equation as far as I can see. That's what is bothering me what can I do about it?

$$Y(s)s^2 + Y(s)2s + Y(s) = e^{-s} - Y(s)9$$
____

$$Y(s)((s+1)^2 +9) = e^{-s}$$

6. Oct 14, 2009

### monet A

Thanks for your help Rock. I have discovered a table which answers my question so just ignore the recent post, I'm okay now. :)