Evaluate the inverse Laplace transform of the function 2s+3/2s-3 for 2 marks answer

To find the inverse Laplace transform of the function \(\frac{2s+3}{2s-3}\), we can rewrite it in a form that is easier to transform back to the time domain. 

The given function is:

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

### Step 1: Simplify the Expression

First, let's simplify the expression by dividing each term in the numerator by the denominator:

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

Simplifying further:

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

### Step 2: Inverse Laplace Transform

Now, let's find the inverse Laplace transform of each term separately.

1. **Inverse Laplace Transform of 1:**

   The inverse Laplace transform of \(1\) is the Dirac delta function at time zero:

   \[
   \mathcal{L}^{-1}\{1\} = \delta(t)
   \]

2. **Inverse Laplace Transform of \(\frac{6}{2s - 3}\):**

   To find the inverse Laplace transform of this term, note that it can be written as a constant times the standard form of \(\frac{1}{s-a}\), which has the inverse Laplace transform \(e^{at}\):

   \[
   \frac{6}{2s - 3} = 3 \cdot \frac{2}{2s - 3}
   \]

   Rewriting:

   \[
   \frac{6}{2s - 3} = 3 \cdot \frac{1}{s - \frac{3}{2}}
   \]

   The inverse Laplace transform of \(\frac{1}{s-a}\) is \(e^{at}\):

   \[
   \mathcal{L}^{-1}\left\{\frac{1}{s - \frac{3}{2}}\right\} = e^{\frac{3}{2}t}
   \]

   Therefore, the inverse Laplace transform of \(3 \cdot \frac{1}{s - \frac{3}{2}}\) is:

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

### Final Result

Combining these results, the inverse Laplace transform of \(F(s)\) is:

\[
f(t) = \delta(t) + 3e^{\frac{3}{2}t}
\]

Thus, the inverse Laplace transform of \(\frac{2s+3}{2s-3}\) is:

\[
f(t) = \delta(t) + 3e^{\frac{3}{2}t}
\]

This function represents a combination of an impulse at \(t = 0\) and an exponentially growing function for \(t > 0\).

---

If you need further details or have any questions, feel free to ask!

Here are some related questions you might find interesting:

1. What is the inverse Laplace transform of \(\frac{s}{s^2 + 4}\)?
2. How do you perform partial fraction decomposition for Laplace transforms?
3. What is the Laplace transform of \(e^{at}\sin(bt)\)?
4. Can you explain the use of convolution in Laplace transforms?
5. How does the Laplace transform apply to solving differential equations?

**Tip:** When solving inverse Laplace transform problems, try to express the function in terms of standard forms for which you know the inverse directly from Laplace transform tables.

Learn how to evaluate the inverse Laplace transform of the function (2s + 3) / (2s - 3). This problem covers the application of Laplace transform properties and formulas, suitable for university-level students. Includes step-by-step solution and explanations.

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