Consider the first-order homogeneous system of linear differential equations

What is the general solution of this system? Here, and are arbitrary real constants.
A) $ \mathbf{x}(t)=C_{1} e^{-2 t}\left[\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{-2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $
B) $ \mathbf{x}(t)=C_{1} e^{8 t}\left(\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right)+C_{2} e^{8 t}\left(\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $
C)$ \mathbf{x}(t)=C_{1} e^{-8 t}\left[\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right]+C_{2} e^{-8 t}\left[\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $
D) $ \mathbf{x}(t)=C_{1} e^{2 t}\left(\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $

Question

Consider the first-order homogeneous system of linear differential equations
 
What is the general solution of this system? Here,  and  are arbitrary real constants.
A) $ \mathbf{x}(t)=C_{1} e^{-2 t}\left[\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{-2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $ 
B) $ \mathbf{x}(t)=C_{1} e^{8 t}\left(\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right)+C_{2} e^{8 t}\left(\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $ 
C)$ \mathbf{x}(t)=C_{1} e^{-8 t}\left[\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right]+C_{2} e^{-8 t}\left[\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $ 
D) $ \mathbf{x}(t)=C_{1} e^{2 t}\left(\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $

Quizplus · Accepted Answer

The Answer of Consider the first-order homogeneous system of linear differential equations
 
What is the general solution of this system? Here,  and  are arbitrary real constants.
A) $ \mathbf{x}(t)=C_{1} e^{-2 t}\left[\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{-2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $ 
B) $ \mathbf{x}(t)=C_{1} e^{8 t}\left(\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right)+C_{2} e^{8 t}\left(\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $ 
C)$ \mathbf{x}(t)=C_{1} e^{-8 t}\left[\begin{array}{l}\sin (2 t) \ \cos (2 t)\end{array}\right]+C_{2} e^{-8 t}\left[\begin{array}{l}-\cos (2 t) \ \sin (2 t)\end{array}\right) $ 
D) $ \mathbf{x}(t)=C_{1} e^{2 t}\left(\begin{array}{l}\sin (-8 t) \ \cos (-8 t)\end{array}\right)+C_{2} e^{2 t}\left[\begin{array}{l}-\cos (-8 t) \ \sin (-8 t)\end{array}\right) $

Consider the First-Order Homogeneous System of Linear Differential Equations