2.3.2.5 ✓ Mathematica. Time used: 0.027 (sec). Leaf size: 27

ode=D[x[t],{t,2}]-x[t]==Exp[t]; 
ic={x[0]==0,Derivative[1][x][0] == 1}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {1}{4} e^{-t} \left (e^{2 t} (2 t+1)-1\right ) \end{align*}