Inverting a Laplace Transform w/non-factorable denominator

[monet A]

Homework Statement

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

Homework 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.

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.

 

[rock.freak667]

try completing the square for s²+2s+10

 

[monet A]

try completing the square for s²+2s+10

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?

 

[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?}

 

[monet A]

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?}

<sup>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}</sup>

 

[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. :)