Find $y(5)$

$\ddot{y} + \dot{y} - 6y = 1$

Taking Laplace transform on both sides:

$s^2Y(s) - sy(0) - \dot{y}(0) + sY(s) - y(0) - 6Y(s) = \frac{1}{s}$

$\because y(0) = \dot{y}(0) = 0$ $\implies (s^2 + s - 6)Y(s) = \frac{1}{s}$ $\implies Y(s) = \frac{1}{s(s^2+s-6)}$ $\implies Y(s) = \frac{1}{s(s-2)(s+3)}$ After splitting into partial fractions and simplifying:

$Y(s) = \frac{1}{10(s-2)} - \frac{1}{6s} + \frac{1}{15(s+3)}$

By performing the inverse Laplace transform on both sides leads to:

$y(t) = \frac{\mathrm{e}^{2t}}{10} - \frac{1}{6} + \frac{\mathrm{e}^{-3t}}{15}$ $\therefore \boxed{y(5) = \frac{\mathrm{e}^{10}}{10} - \frac{1}{6} + \frac{\mathrm{e}^{-15}}{15} \approx 2202.48}$