Consider the first-order homogeneous system of linear differential equations

The eigenvalues and corresponding eigenvectors for this system are:

Which of these is the general solution for this system?
A)$ \mathbf{x}(t)=C_{1} e^{t}\left[\begin{array}{l}\sin (5 t) \ -\cos (5 t)\end{array}\right)+C_{2} e^{t}\left[\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right) $
B) $ \mathbf{x}(t)=C_{1} e^{t}\left(\begin{array}{l}\sin (5 t) \ \cos (5 t)\end{array}\right)+C_{2} e^{t}\left(\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right) $
C) $ \mathbf{x}(t)=C_{1} e^{t}\left[\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right]+C_{2} e^{t}\left[\begin{array}{l}\cos (5 t) \ -\sin (5 t)\end{array}\right) $
D) $ \mathbf{x}(t)=C_{1} e^{t}\left\{\begin{array}{l}\sin (5 t) \ \cos (5 t)\end{array}\right]+C_{2} e^{t}\left[\begin{array}{l}\cos (5 t) \ -\sin (5 t)\end{array}\right] $

Question

Consider the first-order homogeneous system of linear differential equations
 
The eigenvalues and corresponding eigenvectors for this system are:
 
Which of these is the general solution for this system?
A)$ \mathbf{x}(t)=C_{1} e^{t}\left[\begin{array}{l}\sin (5 t) \ -\cos (5 t)\end{array}\right)+C_{2} e^{t}\left[\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right) $ 
B) $ \mathbf{x}(t)=C_{1} e^{t}\left(\begin{array}{l}\sin (5 t) \ \cos (5 t)\end{array}\right)+C_{2} e^{t}\left(\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right) $ 
C) $ \mathbf{x}(t)=C_{1} e^{t}\left[\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right]+C_{2} e^{t}\left[\begin{array}{l}\cos (5 t) \ -\sin (5 t)\end{array}\right) $ 
D) $ \mathbf{x}(t)=C_{1} e^{t}\left\{\begin{array}{l}\sin (5 t) \ \cos (5 t)\end{array}\right]+C_{2} e^{t}\left[\begin{array}{l}\cos (5 t) \ -\sin (5 t)\end{array}\right] $

Quizplus · Accepted Answer

The Answer of Consider the first-order homogeneous system of linear differential equations
 
The eigenvalues and corresponding eigenvectors for this system are:
 
Which of these is the general solution for this system?
A)$ \mathbf{x}(t)=C_{1} e^{t}\left[\begin{array}{l}\sin (5 t) \ -\cos (5 t)\end{array}\right)+C_{2} e^{t}\left[\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right) $ 
B) $ \mathbf{x}(t)=C_{1} e^{t}\left(\begin{array}{l}\sin (5 t) \ \cos (5 t)\end{array}\right)+C_{2} e^{t}\left(\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right) $ 
C) $ \mathbf{x}(t)=C_{1} e^{t}\left[\begin{array}{l}-\sin (5 t) \ \cos (5 t)\end{array}\right]+C_{2} e^{t}\left[\begin{array}{l}\cos (5 t) \ -\sin (5 t)\end{array}\right) $ 
D) $ \mathbf{x}(t)=C_{1} e^{t}\left\{\begin{array}{l}\sin (5 t) \ \cos (5 t)\end{array}\right]+C_{2} e^{t}\left[\begin{array}{l}\cos (5 t) \ -\sin (5 t)\end{array}\right] $

Consider the First-Order Homogeneous System of Linear Differential Equations