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(a) Use Kirchoff's Current Law to derive the single ODE that relates the input voltage V into the output voltage V out in the circuit below.

(b) Use Kirchoff's Current Laws to derive two ODEs that model the electrical circuit to the right. Define vi as the voltage at node 1 and v2 as the voltage at node 2.Write the two ODEs in terms of the state variables x₁ = v₁ and x2 = v2, the control input u = vs(t), and duldt.

(c) Use Kirchoff's Current and Voltage Laws to derive four ODEs that model the circuiton the next page. One equation will be the application of Kirchoff's Current Law at the node between the resistor and the second capacitor (C₂). Reduce these four equations down to three ODEs in terms of the state variables x₁ = VC1, X2 = VC2, X3 = iL and the control input u = vi. Here, vci is the voltage drop across capacitor 1; vc² is the voltage drop across capacitor 2; iL is the current through the inductor; and vi is the voltage supplied by the voltage source.

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