 Solution

Twice integrating by parts, with both endpoint variations fixed, gives $g_x-d(g_{\dot{x}})/dt+d^2(g_{\ddot{x}})/d{t}^2=0$. In the given integral the mixed $\dot{x}\ddot{x}$ term is a boundary term, and the equation is $(D^2-4{)}^{2}x=0$. Decay removes the growing solutions, so $x=(A+Bt)e^{-2t}$. The initial data give $A=1,B=3$, hence $x=(1+3t)e^{-2t}$.