Example: Find the Laplace transform of \(f(t) = 2 + 5t\).
| \(f(t)\) | \(F(s)\) | Region of Application |
|---|---|---|
| \(a\) (constant) | \(\dfrac{a}{s}\) | \(s > 0\) |
| \(t\) | \(\dfrac{1}{s^{2}}\) | \(s > 0\) |
\[\mathcal{L}\{2 + 5t\} = 2\,\mathcal{L}\{1\} + 5\,\mathcal{L}\{t\}\]
\[= \; 2\,\frac{1}{s} + 5\,\frac{1}{s^{2}} = \frac{2}{s} + \frac{5}{s^{2}}, \qquad s > 0.\]
Notes: We used linearity of the Laplace transform (\(\mathcal{L}\{a f + b g\} = a\,\mathcal{L}\{f\} + b\,\mathcal{L}\{g\}\)), and the standard transforms
\(\mathcal{L}\{1\} = 1/s\) and \(\mathcal{L}\{t\} = 1/s^2\). To typeset inline math, wrap your LaTeX in \\( ... \\); for display math use \\[ ... \\].