Given ( F(s) = \frac{4s + 6}{s^2(s+1)} ), find the partial fraction decomposition laplace transform

To find the partial fraction decomposition of the given Laplace transform \( F(s) = \frac{4s + 6}{s^2(s+1)} \), we need to express it in a form where we can easily apply the inverse Laplace transform. The general form of partial fractions for a function of this type is:

\[
F(s) = \frac{4s + 6}{s^2(s+1)} = \frac{A}{s} + \frac{B}{s^2} + \frac{C}{s+1}
\]

Here, we will find the constants \( A \), \( B \), and \( C \).

### Step 1: Set up the equation

\[
\frac{4s + 6}{s^2(s+1)} = \frac{A}{s} + \frac{B}{s^2} + \frac{C}{s+1}
\]

Multiply both sides by \( s^2(s+1) \) to eliminate the denominators:

\[
4s + 6 = A s(s+1) + B(s+1) + C s^2
\]

### Step 2: Expand the terms

Expand both sides of the equation:

\[
4s + 6 = A s^2 + A s + B s + B + C s^2
\]

Group like terms:

\[
4s + 6 = (A + C) s^2 + (A + B) s + B
\]

### Step 3: Equate coefficients

Now, equate the coefficients of the powers of \( s \) on both sides:

- For \( s^2 \): \( A + C = 0 \)
- For \( s \): \( A + B = 4 \)
- For the constant term: \( B = 6 \)

### Step 4: Solve the system of equations

From \( B = 6 \), substitute into the second equation:

\[
A + 6 = 4 \implies A = -2
\]

Now substitute \( A = -2 \) into the first equation:

\[
-2 + C = 0 \implies C = 2
\]

### Step 5: Write the partial fraction decomposition

Substitute the values of \( A \), \( B \), and \( C \) into the partial fraction decomposition:

\[
\frac{4s + 6}{s^2(s+1)} = \frac{-2}{s} + \frac{6}{s^2} + \frac{2}{s+1}
\]

Thus, the partial fraction decomposition is:

\[
F(s) = \frac{-2}{s} + \frac{6}{s^2} + \frac{2}{s+1}
\]

### Step 6: Inverse Laplace Transform

To find the inverse Laplace transform, apply the standard transforms:

- \( \mathcal{L}^{-1} \left\{ \frac{-2}{s} \right\} = -2 \)
- \( \mathcal{L}^{-1} \left\{ \frac{6}{s^2} \right\} = 6t \)
- \( \mathcal{L}^{-1} \left\{ \frac{2}{s+1} \right\} = 2e^{-t} \)

Thus, the inverse Laplace transform is:

\[
f(t) = -2 + 6t + 2e^{-t}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you handle repeated factors in partial fraction decomposition?
2. What is the Laplace transform of \( e^{-at} \)?
3. How would the partial fraction decomposition change if the denominator was \( s^2(s+2) \)?
4. Can you apply partial fractions to solve differential equations with Laplace transforms?
5. What is the significance of the terms in the inverse Laplace transform?

### Tip:
When solving for constants in partial fraction decomposition, equate powers of \( s \) carefully to avoid errors in coefficients!

Given ( F(s) = \frac{4s + 6}{s^2(s+1)} ), find the partial fraction decomposition Laplace transform.

Learn how to find the partial fraction decomposition of F(s) = (4s + 6) / s^2(s+1) and apply the inverse Laplace transform. This detailed solution covers decomposition, solving for constants, and using standard Laplace transforms to find f(t). Suitable for university-level calculus or engineering students.

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