Q.3 Circuits, Signals and Systems 2023-2024 The dynamic behaviour of an active electrical circuit that is used as part of a order differential equation, for timer 20, as follows: non-linear temperature

controller in a refinery can be modelled by a first dy(t) dt +Sy(t) = 5x(t) where y(t) is the voltage at the output port and x(t) is the voltage applied to the input terminal at time t= 0 and y(0) = 0. a) Show that the impulse response of the circuit is h(t) = 5e- (5 marks) b) Find the output y(t) when the input is x(t)e-5 by using the following two methods: i. Direct convolution h(t)x(t-1)dr y(t) h(r = (7 marks) ii. Indirect convolution method by using the Laplace transform (7 marks) c) Find the output y(t) when an input function of x (t) = 2u(t) is applied to the system. (6 marks)