Given: A system is linear time invariant with the following information • Zeros: –3 ± j4 • Poles: –1 + j2 • H(0) = 15 Find: • H(w) • Associate differential equation. • Response ofthe system when the input is cos(4t) e-3tu(t)

Please assist with this practice problem 1 c,d,e with details on how to do it. Thank you.

Subject is Signals and Systems can you solve this problem please :)

Charging Capacitor: For a charging capacitor the Kirchoff's Loop Rule gives R E – IR C ww C In this case the current is entering the positive plate so I = dQ/dt = CdAVc/dt and we get dVc E – RC - Vc = 0 dt The solution to the differential equation is V.(t) = E (1 – e-t/(RC)) (charging capacitor) Notice that V.(0) = 0 and V.(0) = E as we expect for a charging capacitor. 5. You can easily find the time constant if you are given a graph of voltage across a charging capacitor as a function of time. Whent = RC, the voltage across the capacitor is V.(t = RC) = E(1 – e-1) × 0.63 E. Therefore the time constant is just how long it takes for AV(t) to reach 63% of the EMF. The graph shows the voltage across a charging capacitor as a function of time. The resistance of the circuit is 7.5 kN. а. Determine the capacitance of the capacitor. 10 8 60 40 time (ms) 20 80 100 b. What is the current at t = 10 ms? Hint: the easiest way to do this is to use the loop rule. (volts) 4.

2. For the following circuit, a. find a. b. find wo. c. write a differential equation in i(t). d. find the characteristic equation. e. find the roots of the characteristic equation. f. find the final value of current i(t) through the inductor. g. find current i(t) through the inductor for t≥ 0 and plot i(t). h. find voltage v(t) for t≥ 0 and plot v(t). 15 mA R 1 k lo-5 mA 100 L 40 mH v(t) - с 50 nF V-2V

(c) A solution of the differential equation dy dy + 2 dx + y = 4e¬x dx2 takes the value 1 when x = O and the value e-1 when x = 1. What is its value when = 2?