# Partial fractions for Laplace transformWatch

#1
Hi,

Is there any other way to find the coefficients for the partial fraction of 8s/((s^+1)(s^2+4s+5)) to find the Laplace transform other than having to plug in 4 values for s and use matrix methods to solve the simultaneous equations? It's the first question in an exercise so I feel that I am maybe missing something quite obvious.

Any help would be appreciated Thank you
0
4 weeks ago
#2
(Original post by Gibbo_)
Hi,

Is there any other way to find the coefficients for the partial fraction of 8s/((s^+1)(s^2+4s+5)) to find the Laplace transform other than having to plug in 4 values for s and use matrix methods to solve the simultaneous equations? It's the first question in an exercise so I feel that I am maybe missing something quite obvious.

Any help would be appreciated Thank you
Nothing appears obvious. Looks like you'll get a couple of sinusoidal terms, an e^-t and an e^-4t.

If you don't factorize the (s^2+4s+5) immediately, you get
(As+B)/(s^2+1) + (Cs+D)/(s^2+4s+5)
You can spot that
D = -5B (constant coefficient)
A = -C (s^3 coefficient
That will reduce it to a couple of simultaneous equations with two variables for the s and s^2, which isn't too bad. Then factorize the (Cs+D)/(s^2+4s+5) part. Note that you're equating coefficients here, rather than subbing values for s.
0
4 weeks ago
#3
(Original post by Gibbo_)
Hi,

Is there any other way to find the coefficients for the partial fraction of 8s/((s^+1)(s^2+4s+5)) to find the Laplace transform other than having to plug in 4 values for s and use matrix methods to solve the simultaneous equations? It's the first question in an exercise so I feel that I am maybe missing something quite obvious.

Any help would be appreciated Thank you

so

When we get: (*)

When , we get:

hence and . Hence and .

Hence (*) implies .

Visually (or, just by inspection), the coeff of on the RHS will be which must be zero when compared to the LHS. Hence .

So without too much work.
Last edited by RDKGames; 4 weeks ago
0
#4
(Original post by mqb2766)
Nothing appears obvious. Looks like you'll get a couple of sinusoidal terms, an e^-t and an e^-4t.

If you don't factorize the (s^2+4s+5) immediately, you get
(As+B)/(s^2+1) + (Cs+D)/(s^2+4s+5)
You can spot that
D = -5B (constant coefficient)
A = -C (s^3 coefficient
That will reduce it to a couple of simultaneous equations with two variables for the s and s^2, which isn't too bad. Then factorize the (Cs+D)/(s^2+4s+5) part. Note that you're equating coefficients here, rather than subbing values for s.
(Original post by RDKGames)

so

When we get: (*)

When , we get:

hence and . Hence and .

Hence (*) implies .

Visually (or, just by inspection), the coeff of on the RHS will be which must be zero when compared to the LHS. Hence .

So without too much work.
Thank you both!
0
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