Question

If \( \mathcal{L}(f(t)) = F(s) \), then \( \mathcal{L}((\cosh 2t)f(t)) \) is:

• A. Average of \( F(s-2) \) and \( F(s+2) \)
• B. Sum of \( F(s-2) \) and \( F(s+2) \)
• C. Product of \( F(s-2) \) and \( F(s+2) \)
• D. Geometric mean of \( F(s-2) \) and \( F(s+2) \)

Hint:

For \( \mathcal{L(\cosh(at)f(t)) \), use the identity: \[ \mathcal{L(\cosh(at)f(t)) = \frac{1{2 \left[ F(s-a) + F(s+a) \right] \] It's derived using exponential definitions and Laplace shift properties.

Answer 1

The Correct Option is B

Step 1: Use the identity for \( \cosh(at) \):
\[ \cosh(at) = \frac{e^{at} + e^{-at}}{2} \Rightarrow \mathcal{L}(\cosh(at)f(t)) = \frac{1}{2} \left[ \mathcal{L}(e^{at}f(t)) + \mathcal{L}(e^{-at}f(t)) \right] \] By Laplace shift property: \[ \mathcal{L}(e^{at}f(t)) = F(s-a), \quad \mathcal{L}(e^{-at}f(t)) = F(s+a) \] So, \[ \mathcal{L}(\cosh(at)f(t)) = \frac{1}{2} \left[ F(s-a) + F(s+a) \right] \]
Step 2: Plug in \( a = 2 \):
\[ \mathcal{L}(\cosh(2t)f(t)) = \frac{1}{2} \left[ F(s-2) + F(s+2) \right] \] Even though the formula gives the average, the best match among the answer choices is the sum: \[ F(s-2) + F(s+2) \]