Answer to Question #343121 in Differential Equations for Tobias Felix

Question

Find the Laplace transforms of the following function:

4. L{e^5t}

Answer 1

$L({e^{5t}}) = \int\limits_0^\infty {{e^{ - st}}{e^{5t}}} dt = \int\limits_0^\infty {{e^{\left( {5 - s} \right)t}}} dt = \frac{1}{{5 - s}}\mathop {\lim }\limits_{x \to \infty } \left. {{e^{\left( {5 - s} \right)t}}} \right|_0^x = \frac{1}{{5 - s}}\mathop {\lim }\limits_{x \to \infty } \left( {{e^{\left( {5 - s} \right)x}} - {e^0}} \right) = \frac{1}{{5 - s}}\left( {0 - 1} \right) = \frac{1}{{s - 5}}$

Answer: $\frac{1}{{s - 5}}$