2.7.6.4 ✓ Mathematica. Time used: 0.137 (sec). Leaf size: 39

ode=D[v[u],{u,2}]-6*D[v[u],u]+13*v[u]==Exp[-2*u]; 
ic={}; 
DSolve[{ode,ic},v[u],u,IncludeSingularSolutions->True]
 

\begin{align*} v(u)&\to \frac {e^{-2 u}}{29}+c_2 e^{3 u} \cos (2 u)+c_1 e^{3 u} \sin (2 u) \end{align*}