Theory: Consider an RLC circuit shown below consisting of an inductor with an inductance of L henry (H), a resistor with a resistance of R ohms (12), and a capacitor with a capacitance of C farads (F) driven by a voltage of E(t) volts (V). Given the voltage drop across the resistor is ER= RI, across the inductor is ELL(dI/dt), and across the capacitor is E= Kirchhoff's Law gives 9 C с 1 L +RI +9=E(t) dl dt dq If we differentiate this equation with respect to time and substitute I = a second-order differential equation dt L d²1 dl +R dt² 71 = dt C + dE dt 1 Problem: Now suppose an RLC circuit with a 2 resistor, a 25F capacitor is driven by the voltage E(t) = 0.1t² V. 25 1 100 we obtain Hinductor, and a = D² I and dl dt d² I i. Using differential operator notation, = DI, write the dt² differential equation associated with this circuit in terms of current I, differential operator D, and time t. ii. Find the roots of the auxiliary polynomial of the corresponding homogeneous equation of I. Enter the roots as a list separated by commas. 71, 72= iii. Find the general solution of the corresponding homogeneous equation (complementary solution) for I. Use A and B for the arbitrary constants. In. (t) = iv. Find a particular solution for I. Where needed, round off all your values to at least five decimal places. 1,(t)= v. Find the general solution for I in terms of t and arbitrary contastands A and B. I(t)=

The differential equation of a series RLC circuit is given by

$L\frac{d^{2}I}{d{t}^2}+R\frac{dI}{dt}+\frac{1}{C}I=\frac{dE}{dt}$;

where

$R=\frac{1}{25}\mathrm{\Omega }.$

$L=\frac{1}{100}H.$

$C=25F.$

$E(t)=0.1t^2$, V.

i. We need to write the given differential equation using differential operator, D and time, t.

ii. We need to find the roots of the auxiliary polynomial of the corresponding homogeneous equation of '$I$'.

iii. We need to find the general solution for the corresponding homogeneous equation of '$I$'.

iv. We need to find the particular solution for $I$.

v. We need to find the general solution of $I$ in terms of time and arbitary constants A, B.