We are given the following system of differential equations:

$\frac{dx}{dt} = 5x - y$ $\frac{dy}{dt} = x + 5y$

Step 1: Eliminate $x$ from the system.

To eliminate $x$, we will differentiate the first equation with respect to $t$. This gives:

$\frac{d^2x}{dt^2} = \frac{d}{dt}(5x - y) = 5\frac{dx}{dt} - \frac{dy}{dt}$

Substitute the expressions for $\frac{dx}{dt}$ and $\frac{dy}{dt}$ from the given system:

$\frac{d^2x}{dt^2} = 5(5x - y) - (x + 5y)$

Simplify:

$\frac{d^2x}{dt^2} = 25x - 5y - x - 5y = 24x - 10y$

Now, use the first equation again to express $x$ in terms of $\frac{dx}{dt}$ and $y$:

$x = \frac{1}{5} \left( \frac{dx}{dt} + y \right)$

Substitute this into the second-order equation:

$\frac{d^2x}{dt^2} = 24 \left( \frac{1}{5} \left( \frac{dx}{dt} + y \right) \right) - 10y$

Simplify:

$\frac{d^2x}{dt^2} = \frac{24}{5} \left( \frac{dx}{dt} + y \right) - 10y$

This simplifies the system to a second-order differential equation for $y$. Solving this yields a solution of the form:

$y(t) = C_1 e^{-5t} \cos(t) + C_2 e^{-5t} \sin(t)$

Conclusion:

Thus, the correct answer is:

A. $y(t) = C_1 e^{-5t} \cos(t) + C_2 e^{-5t} \sin(t)$

Would you like to explore how to find the solution for $x(t)$, or do you have any other questions on this? Here are some related questions for further understanding:

1. How do you solve for $x(t)$ given $y(t)$?
2. How does the method of elimination compare to the method of substitution for solving systems of differential equations?
3. What is the role of the eigenvalues in solving systems like this?
4. How can we verify the solution of a system of differential equations?
5. What happens if the system has degenerate or singular solutions?

Tip: When solving systems of differential equations, always check the consistency of your initial conditions to ensure