differential systems

Question

all the answers for the equation $$\ddot{y}=ye^{t+\dot{y}}$$ can be reached from solving the threee dimension system of $$\dot{X}=F(X)$$ with considering $$y(t)=x_2(t)$$ and $$F\begin{bmatrix}x_1\\x_2\\x_3\end{bmatrix}=\begin{bmatrix}1\\x_1\\x_2e^{x_1+x_3}\end{bmatrix}$$ is this statement true ?

Answer 1

No. You have variables $(t,y(t),\dot y(t))$, so the second component of the vector field $F$ should be $x_3$.