(Solved): Problem 3. Consider the electric circuit shown in Figure 1. Figure 1: An electric circuit. This cir ...

Problem 3. Consider the electric circuit shown in Figure 1. Figure 1: An electric circuit. This circuit is described by the following system of differential equations. $\frac{d}{dt}\left[\begin{array}{c}I\\ V\end{array}\right]=\left[\begin{array}{cc}-\frac{1}{2}& -\frac{1}{8}\\ 2& -\frac{1}{2}\end{array}\right]\left[\begin{array}{c}I\\ V\end{array}\right],$ where $I(t)$ is the current through the inductor and $V(t)$ is the voltage drop across the capacitor. Please answer the following questions. 1 (a) Find the general solution to (1) in terms of real-valued functions. (b) Find $I(t)$ and $V(t)$ corresponding to initial conditions $I(0)=2$ amperes and $V(0)=3$ volts. (c) Find the limiting values of $I(t)$ and $V(t)$ found in (b) as $t\to \infty$. Do these limiting values depend on the initial conditions? (d) Show that $(0,0)$ is the only critical point of (1), and discuss its stability. Compare your results with your answer to (c).