Try:
{x[t], y[t], z[t]} /.
FullSimplify[
dsol = DSolve[
ode = {
x''[t] + t x'[t] + x[t] == z[t],
y''[t] + y'[t] + y[t] == 0,
z''[t] + z'[t] + z[t] == 0},
{x, y, z}, t]];
ode /. dsol // FullSimplify
(* {{True, True, True}} *)
If you solved for {x[t], y[t], z[t]} instead of {x, y, z}, you will have to do some extra work to substitute the solution into the derivatives x'[t], y'[t], and z'[t]. You didn't show what you substituted, so I cannot comment definitively.
