Partial Fraction Decomposition: Y = 2/[s^3 (s + 1)] + 1/(s + 1)

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Mechaman

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I'm having a tough time finishing this partial fraction question. Can someone please explain the next step here?

IMG_20190928_120349.jpg
 
So the constant C=2.
The coefficient of s is 0, ie B + C = 0, therefore B = …… .
Work your way backwards equating the coeffs of s2 and s3 to 0.
 
Harry_the_cat said:
So the constant C=2.
The coefficient of s is 0, ie B + C = 0, therefore B = …… .
Work your way backwards equating the coeffs of s2 and s3 to 0.
Click to expand...
Is the constant c = 2 because, when s is 0, all other values are multiplied by 0 leaving C = 2?

I can't see how b+c = 0

If s is 0 then b+c multiplies by 0, how is it Equal to it?

Can you show where that comes from?
 
It sounds like you may be taking the approach where you just replace s with each zero of the denominator and directly get the coefficients. That doesn't work in this case, where there is a repeated factor. You have to fall back on the general method, which is to equate coefficients of the LHS and RHS. What you have is

2 = (A+D)s^3 + (A+B)s^2 + (B+C)s + C​

which can be seen as

0s^3 + 0s^2 + 0s + 2 = (A+D)s^3 + (A+B)s^2 + (B+C)s + C​

Therefore,

A+D=0​
A+B=0​
B+C=0​
C=2​

and you solve that system.
 
So 2 is C because the 2 is a constant. If it were 2s^2 on the LHS it would be the (A+B) term that would be 2?
 
If ax3 + bx2 +cx + d = 5x3 + 7x2 +3x- 1, then a=5, b =7, c= 3 and d=-1
 
Writing \(\displaystyle \frac{2}{s^3(s+ 1)}= \frac{A}{s}+ \frac{B}{s^2}+ \frac{C}{s^3}+ \frac{D}{s+ 1}\) then multiplying both sides by \(\displaystyle s^3(s+1)\) we have \(\displaystyle 2= As^2(s+ 1)+ Bs(s+ 1)+ C(s+ 1)+ Ds^3\). Taking s= 0, \(\displaystyle 2= C\). Taking s= -1, \(\displaystyle 2= -A\) so A= -2. There are no other values for s that simplify that much but taking s= 1, \(\displaystyle 2= 2A+ 2B+ 2C+ D= -4+ 2B+ 4+ D\) or \(\displaystyle 2B+ D= 2\). Taking s= 2, \(\displaystyle 2= 12A+ 6B+ 3C+ 8D= -24+ 6B+ 6+ 8D\) or \(\displaystyle 6B+ 8D= 20\).

Solve B+ D= 2 and 6B+ 8D= 20 for B and D. Don't forget to add that additional \(\displaystyle \frac{1}{s+ 1}\).
 
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