Question

Verify that the vector X is a solution of the given system. dx dt = 3x − 5y dy dt = 5x − 7y; X = 1 1 e−2t Writing the system in the form X' = AX for some coefficient matrix A, one obtains the following. X' = X For X = 1 1 e−2t, one has X' = AX = . Since the above expressions , X = 1 1 e−2t is a solution of the given system.

Answer 1

X is a solution of the system.

Explanation:

To verify that the vector X is a solution of the given system:

`(dx)/(dt)=3x-5y \\(dy)/(dt)=5x-7y\\X=\left(\begin{array}{c}1&1\end{array}\right)e^(-2t)`

Writing the system in the form X'=AX for some coefficient matrix A, one obtains the following.

`X'=\left(\begin{array}{cc}3&-5\\5&-7\end{array}\right)X`

`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)`

Similarly:

`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)`

Since the above expressions are equal,
`X=\left(\begin{array}{c}1&1\end{array}\right)e^(-2t)` is a solution of the given system.