The problems listed are about finding the Laplace Transform $\mathcal{L}\{f(t)\}$ of the given functions using Theorem 7.2, which likely includes standard formulas for Laplace transforms. Below are detailed solutions for some of the problems:

Example Solutions:

1. Problem 19: $f(t) = 2t^4$ $\mathcal{L}\{t^n\} = \frac{n!}{s^{n+1}}, \quad \mathcal{L}\{2t^4\} = 2 \cdot \frac{4!}{s^5} = \frac{48}{s^5}$

2. Problem 20: $f(t) = t^5$ $\mathcal{L}\{t^5\} = \frac{5!}{s^6} = \frac{120}{s^6}$

3. Problem 21: $f(t) = 4t - 10$ $\mathcal{L}\{4t\} = 4 \cdot \frac{1}{s^2} = \frac{4}{s^2}, \quad \mathcal{L}\{10\} = \frac{10}{s}$ $\mathcal{L}\{4t - 10\} = \frac{4}{s^2} - \frac{10}{s}$

4. Problem 22: $f(t) = 7t + 3$ $\mathcal{L}\{7t\} = 7 \cdot \frac{1}{s^2} = \frac{7}{s^2}, \quad \mathcal{L}\{3\} = \frac{3}{s}$ $\mathcal{L}\{7t + 3\} = \frac{7}{s^2} + \frac{3}{s}$

5. Problem 23: $f(t) = t^2 + 6t - 3$ $\mathcal{L}\{t^2\} = \frac{2!}{s^3} = \frac{2}{s^3}, \quad \mathcal{L}\{6t\} = \frac{6}{s^2}, \quad \mathcal{L}\{-3\} = -\frac{3}{s}$ $\mathcal{L}\{t^2 + 6t - 3\} = \frac{2}{s^3} + \frac{6}{s^2} - \frac{3}{s}$

These examples show the application of standard Laplace Transform formulas for polynomial and constant functions.

Would you like more details on these or help with a specific problem from the list?

Follow-up Questions:

1. What is the Laplace Transform of $\cosh(kt)$?
2. How would you solve the Laplace Transform for a function involving $e^{at}$?
3. Can you derive the Laplace Transform of $\sin(bt)$?
4. What theorem might be used for transforming $(t + 1)^3$?
5. How do we apply the shifting property in the Laplace Transform?

Tip: The Laplace Transform of a polynomial $t^n$ is $\frac{n!}{s^{n+1}}$.