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# -1/(s^2(s + 1)) =􀀀 - 1/s^2 + 1/s 􀀀+ 1/(s+1) Why?

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439
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In my math book it says: $${\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{\left({{\mathtt{s}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}\left({\mathtt{s}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)\right)}}$$=$${\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{\left({\mathtt{s}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}}{\mathtt{\,-\,}}{\frac{{\mathtt{1}}}{\left({{\mathtt{s}}}^{{\mathtt{2}}}\right)}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{s}}}}$$
But I just can't wrap my head around it! How do they get from left to right?
Thanks so much if you can :)

Guest Jun 30, 2015

#2
+85819
+10

This is known as a partial fraction decomposition....let's  look at  this......

-1 / [ s^2 * ( s + 1) ]   ....we can split this up as

-1 / [ s^2 * (s + 1)]  =  A / s   +  B /s^2  + C / (s + 1)    ...where A, B and C are coefficients to be determined

Multiply both sides by [ s^2 * (s + 1)]      ....so we have......

-1 =   As(s + 1) + B (s + 1) + Cs^2        simplify  (factor)  the right side like so .....

-1 = (A + C)s^2 + (A + B)s  + B     then, equating coefficients sets up the following  system

B = -1

A + B =  0    which implies that A = 1

A + C = 0   which implies that C = -1    so we have

-1 / [ s^2 * (s + 1)]  =  1 / s   -  1 /s^2  - 1 / (s + 1)   which is essentially the same as your result

If you're unfamiliar with this technique, here's a good primer........http://www.purplemath.com/modules/partfrac.htm

CPhill  Jun 30, 2015
Sort:

#1
0

"How do they get from left to right?"

You are trying wrong way.. it's simple to get from right to left.

- 1 / (s + 1) - 1 / s^2 + 1/s = - s^2 / (s^2(s+1)) + (- s - 1)/(s^2(s+1)) + (s^2 + s)/(s^2(s+1))

= - 1 / (s^2(s+1))

Guest Jun 30, 2015
#2
+85819
+10

This is known as a partial fraction decomposition....let's  look at  this......

-1 / [ s^2 * ( s + 1) ]   ....we can split this up as

-1 / [ s^2 * (s + 1)]  =  A / s   +  B /s^2  + C / (s + 1)    ...where A, B and C are coefficients to be determined

Multiply both sides by [ s^2 * (s + 1)]      ....so we have......

-1 =   As(s + 1) + B (s + 1) + Cs^2        simplify  (factor)  the right side like so .....

-1 = (A + C)s^2 + (A + B)s  + B     then, equating coefficients sets up the following  system

B = -1

A + B =  0    which implies that A = 1

A + C = 0   which implies that C = -1    so we have

-1 / [ s^2 * (s + 1)]  =  1 / s   -  1 /s^2  - 1 / (s + 1)   which is essentially the same as your result

If you're unfamiliar with this technique, here's a good primer........http://www.purplemath.com/modules/partfrac.htm

CPhill  Jun 30, 2015
#3
+5

Thank you so much! I totally forget about that method, I only think about it while integrating. You've been a great help and brought me one step closer to passing my exam :)

Guest Jun 30, 2015

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