Differential Equations

**Method 1: First Shifting Property** The standard Laplace transform of $\cos(\omega t)$ is: $\mathcal{L}\{\cos(4t)\} = \frac{s}{s^2 + 16}$ Using the first frequency shifting theorem $\mathcal{L}\{e^{at} g(t)\} = G(s - a)$ with $a = 3$: $F(s) = \mathcal{L}\{e^{3t} \cos(4t)\} = \frac{s - 3}{(s - 3)^2 + 16}$ Evaluating at $s = 5$: $F(5) = \frac{5 - 3}{(5 - 3)^2 + 16} = \frac{2}{2^2 + 16} = \frac{2}{4 + 16} = \frac{2}{20} = 0.10$