{x[t],y[t],z[t]}/.FullSimplify[DSolve[
  {x''[t]+t x'[t]+x[t]==z[t],y''[t]+y'[t]+y[t]==0,z''[t]+z'[t]+z[t]==0},
  {x[t],y[t],z[t]},t]]

In[1]:= $VersionNumber

Out[1]= 14.1

In[2]:= {x[t],y[t],z[t]}/.FullSimplify[DSolve[
  {x''[t]+t x'[t]+x[t]==z[t],y''[t]+y'[t]+y[t]==0,z''[t]+z'[t]+z[t]==0},
  {x[t],y[t],z[t]},t]]

Out[2]= {{(6*C[2] + Sqrt[2*Pi]*(E^(1/4 - (I/4)*Sqrt[3])*(Sqrt[3]*E^((I/2)*
  Sqrt[3])*C[4]*Erf[(I + Sqrt[3] - (2*I)*t)/(2*Sqrt[2])] - (3*I)*C[3]*Erf[(-I +
  Sqrt[3] + (2*I)*t)/(2*Sqrt[2])]) + 3*C[1]*Erfi[t/Sqrt[2]]))/(6*E^(t^2/2)),
  (C[6]*Cos[(Sqrt[3]*t)/2] + C[5]*Sin[(Sqrt[3]*t)/2])/E^(t/2), 
  ((-3 - (3*I)*Sqrt[3])*C[3] + (3 + I*Sqrt[3])*E^(I*Sqrt[3]*t)*C[4])/(6*E^(((1 +
  I*Sqrt[3])*t)/2))}}

with no warning or error messages.

But, unless I have made a mistake, substituting the solution into the original three equations does not result in {True,True,True}

It does not work using Mathematica 10.0. Will it work if I use Mathematica 14.1?