Question: (RLC circuit) The circuit in the figure consists of a resistor (R ohms), an inductor (L henrys), a capacitor (C farads), and an initial voltage source. Let b = R/(2L), and suppose R, L, and C have been selected so that b also equals 1/√LC. (This is done, for instance, when the circuit is used in a voltmeter.) Let v (t) be the voltage (in volts) at time t, measured across the capacitor. It can be shown that v is in the null space H of the linear transformation that maps v (t) into Lv"(t) + Rv'(t) + (1/C) v (t), and H consists of all functions of the form v(t) = e -bt (c1 + c2t). Find a basis for H.
