Partial Fraction Decomposition

[icesalmon]

Homework Statement

use partial fraction decomposition to re-write 1/(s²(s²+4)

The Attempt at a Solution

I thought it would break down into (A/s) + (B/s²) + ((cx+d)/(s²+4)
but it doesn't.

 

[Mark44]

Homework Statement

use partial fraction decomposition to re-write 1/(s²(s²+4)

The Attempt at a Solution

I thought it would break down into (A/s) + (B/s²) + ((cx+d)/(s²+4)
but it doesn't.

The only thing wrong is that the last term should be (cs + d)/(s² + 4). Otherwise, your decomposition is correct.

 

[icesalmon]

Okay, thank you sir.

 

[icesalmon]

okay I had this problem solved, but I went back after changing my variable from x to s
and I get A(s)(s²+4) + B(s²+4) +(cs+d)(s²) = 1 (1)
if I let s = 0 then B(4) = 1 -> B = 1/4
if I let s = +/-2i then +/-2iC + D = -1/4 (2)
where as before I equate coefficients it's obvious to me that +/-2iXC != -1/4 and D = -1/4 so C = 0
Letting s = 1 after I get A = 0 also,
my questions are how do I come to the conclusion that C = 0 in (2)? Should I keep plugging values into (1) and try to create a system? Also, how do I pick values for A, once I've actually calculated D, C, and B. What would cause me to choose 1 for S as opposed to any other value?

 

[Ray Vickson]

Homework Statement

use partial fraction decomposition to re-write 1/(s²(s²+4)

The Attempt at a Solution

I thought it would break down into (A/s) + (B/s²) + ((cx+d)/(s²+4)
but it doesn't.

Just write $s^2 = x$, so you have
$$\frac{1}{x(x+4)}$$
The partial fraction expansion for this is
$$\frac{1}{4x} - \frac{1}{4(x+4)},$$
and you can now put back $x = s^2$ to get
$$\frac{1}{4 s^2} - \frac{1}{4(s^2+4)}$$
If you want, you can even introduce complex numbers and bread this down further into
$$\frac{1}{4s^2} +\frac{1}{16 i (s - 2i)} - \frac{1}{16 i (s + 2i)}$$
where $i = \sqrt{-1}$.

 

[icesalmon]

wow, very cool. I've never done it like that before. Thanks a lot

 

[Mark44]

okay I had this problem solved, but I went back after changing my variable from x to s
and I get A(s)(s²+4) + B(s²+4) +(cs+d)(s²) = 1 (1)
if I let s = 0 then B(4) = 1 -> B = 1/4
if I let s = +/-2i then +/-2iC + D = -1/4 (2)
where as before I equate coefficients it's obvious to me that +/-2iXC != -1/4 and D = -1/4 so C = 0
Letting s = 1 after I get A = 0 also,
my questions are how do I come to the conclusion that C = 0 in (2)?

Because -1/4 has no imaginary part. You can think of your equation as being written like this:
D + 2Ci = -1/4 + 0i
From this we see that D = -1/4 and C = 0.

Should I keep plugging values into (1) and try to create a system? Also, how do I pick values for A, once I've actually calculated D, C, and B. What would cause me to choose 1 for S as opposed to any other value?

It doesn't matter what values you choose for s. The only thing to be concerned with is how convenient a particular value is.

The equation you started with -- A/s + B/s² + (Cs + D)/(s² + 4) -- has to be identically true. IOW, it has to be true for all values of s. Any four values you choose will give you four equations for the unknowns A, B, C, and D. The strategy is to pick values so that some of the terms go away, making your task of solving the system easier.