Find the Inverse Laplace Transform

Question

F(s)=s²/ (s+1)(s²+2s+5)

Answer 1

Hi Zapora!

We can solve this problem using partial fraction decomposition and a few standard inverse Laplace transform formulas.

We are given:

F(s) = s² / [(s + 1)(s² + 2s + 5)]

Our goal is to find the inverse Laplace transform, f(t).

Step 1: Set up partial fractions

Since the denominator contains a linear factor and a quadratic factor, we write:

F(s) = A/(s + 1) + (Bs + C)/(s² + 2s + 5)

Multiply both sides by the original denominator:

s² = A(s² + 2s + 5) + (Bs + C)(s + 1)

Expand:

s² = As² + 2As + 5A + Bs² + Bs + Cs + C

Group like terms:

s² = (A + B)s² + (2A + B + C)s + (5A + C)

Now match coefficients:

A + B = 1

2A + B + C = 0

5A + C = 0

Step 2: Solve for A, B, and C

From the third equation:

C = -5A

From the first equation:

B = 1 - A

Substitute into the second equation:

2A + (1 - A) - 5A = 0

1 - 4A = 0

A = 1/4

Now find B:

B = 1 - 1/4 = 3/4

Find C:

C = -5(1/4) = -5/4

Therefore:

F(s) = 1/[4(s + 1)] + [(3/4)s - 5/4]/(s² + 2s + 5)

Step 3: Complete the square

Look at the quadratic denominator:

s² + 2s + 5

Complete the square:

s² + 2s + 5 = (s + 1)² + 4

We now rewrite the numerator in terms of (s + 1):

(3/4)s - 5/4

= (3/4)(s + 1) - 2

Therefore:

F(s) = 1/[4(s + 1)] + [(3/4)(s + 1) - 2]/[(s + 1)² + 4]

Separate the fractions:

F(s) = 1/[4(s + 1)] + (3/4)(s + 1)/[(s + 1)² + 4] - 2/[(s + 1)² + 4]

Step 4: Apply inverse Laplace transforms

Recall these standard formulas:

1. L⁻¹{1/(s + a)} = e^(-at)
2. L⁻¹{(s + a)/[(s + a)² + b²]} = e^(-at)cos(bt)
3. L⁻¹{b/[(s + a)² + b²]} = e^(-at)sin(bt)

Apply these formulas to each term.

First term:

L⁻¹{1/[4(s + 1)]}

= (1/4)e^(-t)

Second term:

L⁻¹{(3/4)(s + 1)/[(s + 1)² + 4]}

= (3/4)e^(-t)cos(2t)

Third term:

L⁻¹{-2/[(s + 1)² + 4]}

= -e^(-t)sin(2t)

Final Answer

f(t) = (1/4)e^(-t) + (3/4)e^(-t)cos(2t) - e^(-t)sin(2t)

We can also factor out e^(-t):

f(t) = e^(-t)[1/4 + (3/4)cos(2t) - sin(2t)]

Both forms are correct!

Key Takeaway

The trick to problems like this is to:

1. Break the rational expression into partial fractions.
2. Complete the square in the quadratic denominator.
3. Rewrite the numerator to match standard Laplace transform formulas.
4. Apply the inverse Laplace transform to each term.

I hope this helps you understand the process!