How to Do Symbolic and Numerical Laplace Transforms in MATLAB?

Use laplace() and ilaplace() functions from Symbolic Math Toolbox:

syms t s a

% 1. Forward Laplace Transform: f(t) -> F(s)
f = exp(-a*t) * sin(3*t);
F = laplace(f, t, s);
disp('Laplace Transform F(s):');
pretty(F)

% 2. Inverse Laplace Transform: F(s) -> f(t)
F_sys = 1 / (s^2 + 4*s + 13);
f_time = ilaplace(F_sys, s, t);
disp('Inverse Laplace Transform f(t):');
pretty(f_time)