To solve this system of linear differential equations using the elimination method, we will:

1. Write the system of equations: $\frac{dx}{dt} = 5x - y$ $\frac{dy}{dt} = 4x + 5y$

2. Differentiate the first equation with respect to $t$: $\frac{d^2x}{dt^2} = \frac{d}{dt}(5x - y) = 5\frac{dx}{dt} - \frac{dy}{dt}.$ Substitute $\frac{dx}{dt}$ and $\frac{dy}{dt}$ from the original system: $\frac{d^2x}{dt^2} = 5(5x - y) - (4x + 5y).$ Simplify: $\frac{d^2x}{dt^2} = 25x - 5y - 4x - 5y = 21x - 10y.$

3. Eliminate $y$ using the first equation: From $\frac{dx}{dt} = 5x - y$, solve for $y$: $y = 5x - \frac{dx}{dt}.$ Substitute $y$ into the second-order equation: $\frac{d^2x}{dt^2} = 21x - 10(5x - \frac{dx}{dt}).$ Simplify: $\frac{d^2x}{dt^2} = 21x - 50x + 10\frac{dx}{dt}.$ $\frac{d^2x}{dt^2} - 10\frac{dx}{dt} + 29x = 0.$

4. Solve the characteristic equation: The characteristic equation is: $r^2 - 10r + 29 = 0.$ Solve using the quadratic formula: $r = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(29)}}{2(1)} = \frac{10 \pm \sqrt{100 - 116}}{2}.$ $r = \frac{10 \pm \sqrt{-16}}{2} = \frac{10 \pm 4i}{2}.$ $r = 5 \pm 2i.$

5. Write the general solution for $x(t)$: The solution is: $x(t) = e^{5t}(C_1\cos(2t) + C_2\sin(2t)).$

6. Solve for $y(t)$: Substitute $x(t)$ into $y = 5x - \frac{dx}{dt}$. After calculations, the solution for $y(t)$ is: $y(t) = e^{5t}(C_1\cos(2t) + C_2\sin(2t)).$

The correct answer is: D. $y(t) = C_1 e^{5t} \cos(2t) + C_2 e^{5t} \sin(2t)$.

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Related Questions:

1. How can you verify the solution of a system of differential equations?
2. What does the characteristic equation tell us about the behavior of the solutions?
3. Why do we see oscillatory behavior in solutions involving complex eigenvalues?
4. How does the elimination method differ from matrix-based methods for solving linear systems?
5. What physical systems might correspond to this type of differential equation?

Tip:

For second-order differential equations with complex roots, the solution always includes exponential growth/decay combined with sinusoidal functions.