Consider the First-Order Homogeneous System of Linear Differential Equations

Multiple Choice

Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system? Here, is an arbitrary constant vector.

A) $\mathbf{x}(t) =\left(\begin{array}{lll}e^{t} & e^{4 t} & \left(t-\frac{1}{6}\right) e^{4 t} \\ 6 e^{t} & -3 e^{4 t} & -3 t e^{4 t} \\ 3 e^{t} & 0 & 0\end{array}\right) \mathbf{C}$
B) $\mathbf{x}(t) =\left(\begin{array}{lll}e^{t} & e^{4 t} & \left(t+\frac{1}{6}\right) e^{4 t} \\ 6 e^{t} & 3 e^{4 t} & 3 t e^{4 t} \\ 3 e^{t} & 0 & 0\end{array}\right)$ C
C) $\mathbf{x}(t) =\left[\begin{array}{lll}e^{t} & e^{4 t} & \left(t+\frac{1}{6}\right) e^{4 t} \\ -3 e^{t} & -3 e^{4 t} & 3 t e^{4 t} \\ 6 e^{t} & 0 & 0\end{array}\right]$ C
D) $\mathbf{x}(t) =\left[\begin{array}{lll}e^{-t} & e^{-4 t} & \left(t+\frac{1}{6}\right) e^{-4 t} \\ 6 e^{-t} & 3 e^{-4 t} & 3 t e^{-4 t} \\ 3 e^{-t} & 0 & 0\end{array}\right]$ C