Unless you are solving a partial differential equation, such that the Laplace transform produces an ordinary differential equation in one of the two variables and a Laplace transform of ‘t’, dsolve is not appropriate. It is simply necessary to solve for (in this instance) ‘Y(s)’ and then invert it to get ‘y(t)’:
syms t y(t) Y(s) Dy0
Dy2 = diff(y,t,2);
Dy1 = diff(y,t,1);
eqn = laplace(Dy2+3*Dy1+2*y == exp(-t))
eqn = subs(eqn, {laplace(y(t), t, s), subs(diff(y(t), t), t, 0)},{Y(s), Dy0})
cond = [y(0)==4, Dy1(0)==5];
Soln_s = isolate(eqn, Y(s))
Soln_t = ilaplace(Soln_s)
Soln_t = subs(Soln_t, {ilaplace(Y(s), s, t)}, {y(t)})
Soln_t = simplify(Soln_t, 'Steps',500)
.
