3.An RLC Circuit is governed by the differential equationd21R dl+I-0dt2L dtLCwhere I(t) gives the current as a function of t, R is the resistance, L is the inductance, and Cis the capacitance (R2 0, L > 0, C> 0). Without knowing about circuits, you can answerthe following questions, because the structure is the same as for the linear pendulum model!(a.)Suppose here that R 0. Use the idea of the characteristic polynomial to find thegeneral solution formula for I(t).What is the period of the resulting oscillations in current from part (a.)?(b.)Now consider the full differential equation:(c.)d21R dl+dt2+I-0L dtLCWrite this as a first order system of equations. Use the Jacobian matrix to find thecharacteristic equation. Confirm that this is the same characteristic equation as youwould get directly from the second order differential equation.

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Asked Nov 6, 2019
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3.
An RLC Circuit is governed by the differential equation
d21
R dl
+I-0
dt2
L dt
LC
where I(t) gives the current as a function of t, R is the resistance, L is the inductance, and C
is the capacitance (R2 0, L > 0, C> 0). Without knowing about circuits, you can answer
the following questions, because the structure is the same as for the linear pendulum model!
(a.)
Suppose here that R 0. Use the idea of the characteristic polynomial to find the
general solution formula for I(t).
What is the period of the resulting oscillations in current from part (a.)?
(b.)
Now consider the full differential equation:
(c.)
d21
R dl
+
dt2
+I-0
L dt
LC
Write this as a first order system of equations. Use the Jacobian matrix to find the
characteristic equation. Confirm that this is the same characteristic equation as you
would get directly from the second order differential equation.
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3. An RLC Circuit is governed by the differential equation d21 R dl +I-0 dt2 L dt LC where I(t) gives the current as a function of t, R is the resistance, L is the inductance, and C is the capacitance (R2 0, L > 0, C> 0). Without knowing about circuits, you can answer the following questions, because the structure is the same as for the linear pendulum model! (a.) Suppose here that R 0. Use the idea of the characteristic polynomial to find the general solution formula for I(t). What is the period of the resulting oscillations in current from part (a.)? (b.) Now consider the full differential equation: (c.) d21 R dl + dt2 +I-0 L dt LC Write this as a first order system of equations. Use the Jacobian matrix to find the characteristic equation. Confirm that this is the same characteristic equation as you would get directly from the second order differential equation.

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Expert Answer

Step 1

(a)Write the given differential equation and plug 0 for R, we get

d21
R dI
1
I 0
LC
dt
L dt
d2I (0) d
dr
I
1
-I = 0
+
L dt LC
d21
1
I 0
LC
d21
+I0
dr
(Here, o, 1/NLC
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d21 R dI 1 I 0 LC dt L dt d2I (0) d dr I 1 -I = 0 + L dt LC d21 1 I 0 LC d21 +I0 dr (Here, o, 1/NLC

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Step 2

The characteristic equation can be given as,

2
= 0
= tia
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2 = 0 = tia

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Step 3

Hence, the general solution...

I(r) Acos tBsin ot
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I(r) Acos tBsin ot

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