Partial fraction (tricky stuff for LaPlace transforms)

[thomas49th]

Homework Statement

Doing some Laplace transform stuffs and I've got Y(s) = \frac{1}{(s+1)(s^{2}+2s+2)}

Using the normal method

1 = A((s+1)^{2}+1) + B(s+1)

I'm not sure this method is valid though as we had a complicated term (s+1)² + 1 after A. I can find A to be 1, but I don't trust my value of B being -1.

Apparently the answer is \frac{1}{s+1} - \frac{s+1}{(s+1)^{2}+1}

Not sure how to get there though

Thanks
Thomas

 

[Dickfore]

The discrimant of the quadratic trinomial is:

<br /> \Delta = 1^{2} - 1 \cdot 2 = 1 - 2 = -1 &lt; 0<br />

so it does not have real zeros. That is why the partial fraction expansion is of the form:

<br /> \frac{1}{(s + 1)(s^{2} + 2 s + 2)} = \frac{A}{s + 1} + \frac{B s + C}{s^{2} + 2 s + 2}<br />

Multiply both sides by (s + 1)(s^{2} + 2 s + 2), collect terms with like powers of s on the right hand side and compare the corresponding coefficients so that you will get three equations for the three unknowns A, B and C. Also, complete the square for the trinomial:

<br /> s^{2} + 2 s + 2<br />

You should get the answer you were looking for.