Apply the Laplace transform: (Look up in table of Laplace transforms)
\(s L(s) - x_0 + 7 L(s) = \frac{5s}{s^2 +4}\)
Isolate and solve for L(s)
\(L(s) = \frac{x_0}{s+7} + \frac{5s}{(s+7)(s^2+4)}\)
The 2nd term is too complex as is to perform inverse Transform, use partial fraction decomposition is split it up into simpler terms.
\(L(s) = \frac{x_0}{s+7} -\frac{35}{53} (\frac{1}{s+7}) + \frac{35}{53} (\frac{s}{s^2+4})+\frac{10}{53} (\frac{2}{s^2+4})\)
Now apply inverse Laplace Transform using Table:
\(x(t) = C e^{-7t} + \frac{35}{53} cos (2t) + \frac{10}{53} sin(2t)\)
where \(C = x_0 - \frac{35}{53}\)
