+0

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

0
287
3

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
+78750
+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
+78750
+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

11 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details