Partial Fraction Decomposition for (s^2+1)^2 - Homework Help

[wavingerwin]

Homework Statement

How to break down:
$$\frac{1}{(s^{2}+1)^{2}}$$
into partial fractions?

Homework Equations

-

The Attempt at a Solution

I have tried:
$$\frac{1}{(s^{2}+1)^{2}} = \frac{1}{(1+i)^{2}\times(1-i)^{2}} = \frac{A}{(s+i)} + \frac{B}{(s+i)^{2}} + \frac{C}{(s-i)}} + \frac{D}{(s-i)^{2}}$$

and

$$\frac{1}{(s^{2}+1)^{2}} = \frac{As+B}{(s^{2}+1)^{2}} + \frac{Cs+D}{(s^{2}+1)}$$

and

$$\frac{1}{(s^{2}+1)^{2}} = \frac{As+B}{(s^{2}+1)} + \frac{Cs+D}{(s^{2}+1)}$$

but none of them works..
Please help

 

[tiny-tim]

Hi v_bachtiar!

Why isn't 1/(s² + 1)²) good enough as it is?

But if you do want to break it down further, your first try should have worked …

what did you get? ​

 

[wavingerwin]

It is not good enough because I need to perform an inverse Laplace transform on the fraction.
And at my level, I only use tables and some basic theorems (convolution, shift in s etc.) and 1/(s2 + 1)2) is not on the table :(

I have attached my working using the first try..

 

[tiny-tim]

Hi v_bachtiar!

What makes you think A B C and D are real?

Hint: to simplify it, what are 1/(s - i) ± 1/(s + i) and 1/(s - i)² ± 1/(s + i)² ?

 

[wavingerwin]

They are 2s/(s²+1) or 0
and
(2s²-2)/(s²+1)² or 4si/(s²+1)²

so..

(As-Ai+Cs+Ci) / (s²+1) + (B(s+i)²+D(s-i)²) / ((s²+1)²) = 1

and As-Ai+Cs+Ci = 2s , B(s+i)²+D(s-i)² = 2s²-2

is this right?

 

[wavingerwin]

oh, i mean:

(As-Ai+Cs+Ci) + (B(s+i)²+D(s-i)2) = 1

 

[tiny-tim]

Hi v_bachtiar!

(just got up :zzz: …)

They are 2s/(s²+1) or 0
and …

No, 2s/(s²+1) or 2i/(s²+1)

ok, rewrite them as

(2s³+s)/(s²+1)² or i(2s²+2)/(s²+1)²

Now can you see how to easily combine them with the others to get 1/(s²+1)² ?

 

[pradeeppankaj]

Homework Statement

How to break down:
$$\frac{1}{(s^{2}+1)^{2}}$$
into partial fractions?

Homework Equations

-

The Attempt at a Solution

I have tried:
$$\frac{1}{(s^{2}+1)^{2}} = \frac{1}{(1+i)^{2}\times(1-i)^{2}} = \frac{A}{(s+i)} + \frac{B}{(s+i)^{2}} + \frac{C}{(s-i)}} + \frac{D}{(s-i)^{2}}$$

and

$$\frac{1}{(s^{2}+1)^{2}} = \frac{As+B}{(s^{2}+1)^{2}} + \frac{Cs+D}{(s^{2}+1)}$$

and

$$\frac{1}{(s^{2}+1)^{2}} = \frac{As+B}{(s^{2}+1)} + \frac{Cs+D}{(s^{2}+1)}$$

but none of them works..
Please help

u can do as
A/s^2+1 + BX/(S^2+1)2

 

[wavingerwin]

Hi v_bachtiar!

(just got up :zzz: …)


No, 2s/(s²+1) or 2i/(s²+1)

ok, rewrite them as

(2s³+s)/(s²+1)² or i(2s²+2)/(s²+1)²

Now can you see how to easily combine them with the others to get 1/(s²+1)² ?

hi tiny-tim,

you mean combine (add) them with (2s²-2)/(s²+1)²?

so 1 = (2s³+s) + (2s²-2)

then, where do I imply the A, B, C, and D?

(thank you for your help so far)

 

[tiny-tim]

How about (2s²+2) and (2s²-2) ?