Answer:
X is a solution of the system.
Step-by-step explanation:
To verify that the vector X is a solution of the given system:
[tex]\frac{dx}{dt}=3x-5y \\\frac{dy}{dt}=5x-7y\\X=\left(\begin{array}{c}1&1\end{array}\right)e^{-2t}[/tex]
Writing the system in the form X'=AX for some coefficient matrix A, one obtains the following.
[tex]X'=\left(\begin{array}{cc}3&-5\\5&-7\end{array}\right)X[/tex]
[tex]For\:X=\left(\begin{array}{c}1&1\end{array}\right)e^{-2t} , X'=\left(\begin{array}{c}-2&-2\end{array}\right)e^{-2t}[/tex]
Similarly:
[tex]AX=\left(\begin{array}{cc}3&-5\\5&-7\end{array}\right)\left(\begin{array}{c}1&1\end{array}\right)e^{-2t}=\left(\begin{array}{c}-2&-2\end{array}\right)e^{-2t}[/tex]
Since the above expressions are equal, [tex]X=\left(\begin{array}{c}1&1\end{array}\right)e^{-2t}[/tex] is a solution of the given system.
