Factoring in Inverse Laplace Transformations

[xzibition8612]

Homework Statement

see attachment

Homework Equations

The Attempt at a Solution

I don't get the part where it says "Solving this equation for X(s), we obtain ..."
Specifically jumping from 3/s(s²+3s+5) to (3/5)(1/s)-(3/5)[(s+2)/(s²+2s+5)].

How did the problem break up this fraction into separate subtracting fractions?? Is there a trick or something? I'm completely lost here. Thanks.

 

[Ray Vickson]

Homework Statement

see attachment

Homework Equations

The Attempt at a Solution

I don't get the part where it says "Solving this equation for X(s), we obtain ..."
Specifically jumping from 3/s(s²+3s+5) to (3/5)(1/s)-(3/5)[(s+2)/(s²+2s+5)].

How did the problem break up this fraction into separate subtracting fractions?? Is there a trick or something? I'm completely lost here. Thanks.

It is just partial fractions. However, the expression given above is wrong: Maple 11 gets:

gs:=3/s/(s^2+3*s+5):lprint(%);
3/s/(s^2+3*s+5)

(Maple uses notation A/B/C to stand for A/(B*C), etc, and uses A/B*C to mean (A/B)*C, etc.)

g1:=convert(gs,parfrac,s): lprint(%);
3/5/s + 1/5*(-3*s-9)/(s^2+3*s+5)

So, Maple gets -(3/5)(s+3) instead of -(3/5)(s+2) in the numerator of the second term. You can easily check that (s+2) gives the wrong answer when you combine the two terms.

BTW: "partial fractions" is a standard tool in calculus, often used in computing integrals. Just Google 'partial fractions' if you have not seen this material before, or consult a calculus textbook.

 

[xzibition8612]

i get it thx