https://file.aiquickdraw.com/mathbot/file/1730670635873c66cwbcv.jpg

Let's go through the solutions for the inverse Laplace transforms in parts (a) and (b).

### Problem (a)
Given:
\[
F(s) = \frac{2s}{s^2 - 6s + 5}
\]
To find \( f(t) \), we need to compute the inverse Laplace transform of \( F(s) \).

#### Step 1: Factor the Denominator
The denominator is:
\[
s^2 - 6s + 5 = (s - 5)(s - 1)
\]

#### Step 2: Partial Fraction Decomposition
Rewrite \( F(s) \) as:
\[
\frac{2s}{(s - 5)(s - 1)} = \frac{A}{s - 5} + \frac{B}{s - 1}
\]
To solve for \( A \) and \( B \), multiply through by the denominator:
\[
2s = A(s - 1) + B(s - 5)
\]
Expanding and collecting terms gives:
\[
2s = As - A + Bs - 5B
\]
\[
2s = (A + B)s - (A + 5B)
\]
Equating coefficients, we get the system:
1. \( A + B = 2 \)
2. \( -A - 5B = 0 \)

Solving these equations, we find:
- \( A = 5 \)
- \( B = -3 \)

Thus:
\[
F(s) = \frac{5}{s - 5} - \frac{3}{s - 1}
\]

#### Step 3: Inverse Laplace Transform
Now apply the inverse Laplace transform to each term:
\[
f(t) = 5e^{5t} - 3e^{t}
\]

This does not match the given answer, so let's double-check calculations or assumptions.

### Problem (b)
Given:
\[
G(s) = \frac{e^{-s}}{s^2 + 4s + 5}
\]

Find the inverse Laplace transform of (a) F(s) = 2s / (s^2 - 6s + 5) and (b) G(s) = e^(-s) / (s^2 + 4s + 5).

Step-by-step solution to find the inverse Laplace transform of F(s) = 2s / (s^2 - 6s + 5) and G(s) = e^(-s) / (s^2 + 4s + 5). This problem involves partial fraction decomposition and application of the Heaviside function. Suitable for university-level students studying calculus and differential equations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation