Laplace transform,partial fraction problem

1. Aug 12, 2013

ttsky

This is more of a pre calc question but it dose however come from diff eqs, just in case I have made fundamental mistakes, i have posted it here. I have been studying this topic for few days by myself, never had any problems with algebra until here.

really appreciate all of your help.

1. The problem statement, all variables and given/known data
problem comes from here.
$y''-2y'-2y=0$ with initial conditions $y(0) = 2 , y'(0) = 0$

I am stuck trying to decompose this line
$\mathcal Y(s) = (2s+4)/(s^2-2s-2)$

2. Relevant equations
Laplace Transforms Tables

3. The attempt at a solution
$y''-2y'-2y=0$

$s^2\mathcal Y(s) - 2s - 2s\mathcal Y(s) -4 -2\mathcal Y(s) = 0$
$(s^2-2s-2)\mathcal Y(s) -2s-4=0$
$\mathcal Y(s) = (2s+4)/(s^2-2s-2)$
and stuck here.. I cant figure out how to decompose last line.

2. Aug 12, 2013

LCKurtz

Complete the square in the denominator: $(s-1)^2-3$. Then write the numerator as $2(s-1)+6$. Does that help?

3. Aug 12, 2013

HallsofIvy

Staff Emeritus
Are you required to use "Laplace Transform"? I have never quite understood why "Laplace Transform" methods are even taught for differential equations! Just writing out the characteristice equation for the given differential equation, $r^2- 2r+ 2= r^2- 2r+ 1+ 1= 0$ gives $r= 1\pm i$ as characteristic solution and so $y(t)= e^{t}(C_1cos(t)+ C_2 sin(t))$ as general solution to the differential equation.

4. Aug 12, 2013

LCKurtz

That's $r^2 -2r -2$, which changes the answer a bit. While I somewhat agree with your sentiments, the transforms are certainly handy for non-homogeneous terms which are piecewise defined, not to mention the usefulness of the transform space in EE applications.

5. Aug 12, 2013

ttsky

I dont see how this came about. can you elaborate?

6. Aug 12, 2013

ttsky

we spent last semester doing just that, I am only studying ahead so I have yet to find out why myself. i have heard it is important for EE students, which is what I am.

7. Aug 12, 2013

LCKurtz

Are you asking how to complete the square in a quadratic? If so, look in any algebra book or look here:

http://en.wikipedia.org/wiki/Completing_the_square

For the second one, just expand it out to see it's the same.

8. Aug 12, 2013

ttsky

thank you for that, this topic opened a can of loop holes in my algebra! really appreciate your help. understand it now!