# Thread: Partial Fraction Decomp Help!

1. ## Partial Fraction Decomp Help!

I'm trying to do a problem that requires me to use the method of Laplace transforms to solve an initial value problem.

Anyway, I'm getting stuck on the Partial Fraction Decomposition Part of the Problem.

What I have is : (x^2+1)/((x^2)(x^2+2s+2))

The original Problem is: Using the method of Laplace Transofmrs, solve the initial value probem x''+2x'+2x = t ; x(0) = 0 x'(0) = 1

2. I wrote that in correctly. What I need Decomposed by Partial Fractions is:

$X =(s^2+1)/((s^2)(s^2+2*s+2))$

3. $\frac{s^2+1}{s^2(s^2+2s+2)} = \frac{A}{s} +\frac{B}{s^2} + \frac{Cs+D}{s^2+2s+2}$

$s^2+1 = As(s^2+2s+2) + B(s^2+2s+2) + (Cs+D)(s^2)$

$s^2+1 = (A+C)s^3 + (2A+B+D)s^2 + (2A+2B)s + 2B$

equating coefficients ...

$2B = 1$

$B = \frac{1}{2}$

$2(A+B) = 0$

$A = -\frac{1}{2}$

$2A+B+D = 1$

$D = \frac{3}{2}
$

$A+C = 0$

$C = \frac{1}{2}$

$\frac{s^2+1}{s^2(s^2+2s+2)} = \frac{1}{2}\left(-\frac{1}{s} +\frac{1}{s^2} + \frac{s+3}{s^2+2s+2}\right)$