(Solved): Consider the following second-order differential equation, which models a damped harmonic oscillat ...

Consider the following second-order differential equation, which models a damped harmonic oscillator. $$\frac{d^{2}y}{d{t}^2}+6\frac{dy}{dt}+7y=0$$ (a) (4 points) Use the "guess and check" method from class to find two different non-equilibrium solutions $$y_1(t)$$ and $$y_2(t)$$. (b) (2 points) Convert the second-order equation into a system of first-order equations, of the form $$\{\begin{array}{l}\frac{dy}{dt}=\\ \frac{dv}{dt}=\end{array}$$ (c) ( 2 points) Find the $$2\times 2$$ matrix $$A$$ such that the system you wrote in part (b) can be rewritten in the form $$\frac{d\mathbf{Y}}{dt}=A\mathbf{Y}$$ where $$\mathbf{Y}=\left[\begin{array}{l}y\\ v\end{array}\right]$$. (d) (2 points) Compute the eigenvalues of $$A$$, and show your work.