To find the Laplace transform \( L\left\{\frac{6}{(s^2-1)(s+2)}\right\} \), we'll first simplify the expression and then apply partial fraction decomposition.

### Step 1: Factor the denominator
The expression is:

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

Factor \( s^2 - 1 \):

\[
s^2 - 1 = (s - 1)(s + 1)
\]

So the expression becomes:

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

### Step 2: Partial fraction decomposition
We express the function as a sum of simpler fractions:

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

Multiplying both sides by the denominator \((s - 1)(s + 1)(s + 2)\) gives:

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

### Step 3: Solve for A, B, and C
To find \( A \), \( B \), and \( C \), we substitute convenient values of \( s \).

1. **Let \( s = 1 \):**

\[
6 = A(1 + 1)(1 + 2) + 0 + 0 \Rightarrow 6 = 6A \Rightarrow A = 1
\]

2. **Let \( s = -1 \):**

\[
6 = 0 + B(-1 - 1)(-1 + 2) + 0 \Rightarrow 6 = 2B \Rightarrow B = 3
\]

3. **Let \( s = -2 \):**

\[
6 = 0 + 0 + C(-2 - 1)(-2 + 1) \Rightarrow 6 = 3C \Rightarrow C = -2
\]

Thus, the decomposition is:

\[
\frac{6}{(s - 1)(s + 1)(s + 2)} = \frac{1}{s - 1} + \frac{3}{s + 1} - \frac{2}{s + 2}
\]

### Step 4: Apply the inverse Laplace transform
Now, we apply the inverse Laplace transform to each term:

\[
L^{-1}\left\{\frac{1}{s - 1}\right\} = e^{t}
\]

\[
L^{-1}\left\{\frac{3}{s + 1}\right\} = 3e^{-t}
\]

\[
L^{-1}\left\{\frac{2}{s + 2}\right\} = 2e^{-2t}
\]

### Final Answer:
Combining these, we get the inverse Laplace transform:

\[
L^{-1}\left\{\frac{6}{(s - 1)(s + 1)(s + 2)}\right\} = e^{t} + 3e^{-t} - 2e^{-2t}
\]

Would you like any further details or have any questions?

Here are 5 related questions you might find interesting:
1. How do you perform partial fraction decomposition with repeated factors?
2. What is the Laplace transform of \( \frac{1}{s^2(s + 1)} \)?
3. How does the initial value theorem apply to Laplace transforms?
4. Can you explain the convolution theorem in the context of Laplace transforms?
5. What are some applications of Laplace transforms in engineering?

**Tip:** When performing partial fraction decomposition, check for repeated roots and handle them appropriately to avoid errors.

Explore how to find the inverse Laplace transform of 6/[(s^2-1)(s+2)] using partial fraction decomposition. This detailed solution covers key concepts in Laplace transforms, including algebraic manipulation and the application of inverse Laplace transforms. Suitable for advanced high school and college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation