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!