What is the inverse Laplace Transform of 1/[(s+1)(s^2 + 1)]?

[Ready2GoXtr]

Homework Statement

Use partial fraction decomposition to find the inverse Laplace Transform.

F(s)= 1/[(s+1)(s^2 + 1)]

Homework Equations

The Attempt at a Solution

1/[(s+1)(s^2 + 1)] = A/(s+1) + (Bs + C)/(s^2 + 1)

1 = A(s^2 + 1) + (Bs + C)(s+1)

I do not know how to solve for A and B or C

 

[jgens]

To solve for A, let s = -1 and go from there.

To solve for B and C, note that: [As² + (A - 1)]/(s + 1) = Bs + C

 

[Ready2GoXtr]

s = -1
1 = A(1 + 1) + B(-1)^2 + B(-1) + C(-1) + C
A = 1/2

I don't understand your next step
do you mean
[(1/2)(-1)^2 + (1/2 - 1)]/(-1 + 1) = B(-1) + C

 

[jgens]

No, I don't mean that. We have that A = 1/2. This means that, (1/2 - s²)/(s + 1) = Bs + C.

Edit: Fixed algebra errors. Wow, really bad algebra on my part!

 

[HallsofIvy]

Or: choose any 3 values for s to get 3 equations in A, B, and C.

For example, choosing, arbitrarily, s= 1, 2, 3 gives:
s=1 2A+ 2B+ 2C= 1
s=2 5A+ 6B+ 3C= 1
s=3 10A+ 12B+ 4C= 1

Or: multiply out the right side and set corresponding coefficients equal.

1 = A(s^2 + 1) + (Bs + C)(s+1)= As^2+ A+ Bs^2+ Bs+ Cs+ C
= (A+ B)s^2+ (B+ C)s+ (A+ C)
0x^2+ 0x+ 1= (A+ B)s^2+ (B+C)s+ (A+ C) so

A+ B= 0, B+ C= 0, A+ C= 1.

You have three unknown numbers, A, B, and C. Any way you can get three equations to solve for them is valid.

 

[Ready2GoXtr]

Thanks for the help guys. I appreciate it!