Find the charge on the capacitor and the current in an LC circuit when L = 0.1 h, C = 0.1 f, E(t) = 100 sin ?t V, q(0) = 0 C, and i(0) = 0 A.

Solution:Step-1:In this problem we need to find the charge on the capacitor and the current in an LC circuit when .Step-2:We know that,………..(1)Where, We know that, the relation between voltage and current in circuit analysis is .represents the voltage drop across the inductor.represents the voltage drop across the resistor.represents the voltage drop across the capacitor.Given,.Substitute these values in equation (1) we get.…………(2).Step-3:Now the auxiliary equation of the differential equation is :Therefore, Therefore, solution of the homogeneous equation is Step-4:We know that, is the particular solution of the differential equation.Differentiate both sides we get.Substitute these values in equation (1) we get.Comparing the coefficients we get..Therefore, particular solution of the differential equation is Step-5:We know that, the solution of the differential equation is Therefore, .step-6:We know that , At initial position t = 0, i(0) = 0 and q(0) = 0,then we get., since cos(0) =1 , and sin(0) =0..Step-7:Therefore, the charge of the capacitor q is: .The current in LC circuit i is: