3. Electrical Oscillators: Consider the electric circuit shown below.

Three capacitors (two with capacitance C, and one with capacitance Cu) are charged with (potentially different) charges q₁, q2, qs and connected with the polarity shown to two inductors of inductance L.In this problem we will explore how to convert the physical properties of this system into a linear map and how, with a strategic choice, we can convert the general linear map into an eigenvalue equation that describes the system. In future homework problems, we will work on solving the eigenvalue equation to determine the natural modes of the system. a) Remembering that the voltage difference across a capacitor and inductor can be written as V=q/C and V=L(dl/dt), respectively (I is the net current through the inductor), derive two equations that relate the charges q: and q2 to the loop currents 1₁, 12, and I. b) Remember that I-dq/dt-q and rewrite the equations so that they now relate q: and qz to q₁ q2 and q c) Using V=q/C, derive a relationship between q₁, q2, and q₁. Take multiple time derivatives to convert this into a relationship between q, q2, and q. Use this result to eliminate q from your expressions for q; and q₂. d) Algebraically rearrange these equations into the following matrix form: \left[\begin{array}{l}

q_{1} \\

q_{2}

\end{array}\right]=M\left[\begin{array}{l}

q_{1} \\

q_{2}

\end{array}\right] where M is a 2 x 2 matrix. \text { e) Show that } M \text { is a linear map from } \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \text {. } f) To proceed, we make the Ansatz (a fancy German word that roughly means "guess solution")that qi and qz will have a time-dependence that is harmonic (sinusoidal). Let q₁ = A exp(iwt) and q2 = Azexp(iwt). Substitute these values into your matrix equation, take the derivatives, and-simplify. Your result should involve the vector (A1,A₂) on both sides of the equation. Note that the form of your result is now a standard eigenvalue equation.