TEAM PROJECT. Complex Method for Particular Solutions. (a) Find a particular solution of

Chapter 2, Problem 2.1.198

(choose chapter or problem)

TEAM PROJECT. Complex Method for Particular Solutions. (a) Find a particular solution of the complex ODE

\(L \tilde{I}^{\prime \prime}+R \tilde{I}^{\prime}+\frac{1}{C} \tilde{I}=E_{0} \omega e^{i \omega t} \quad(i=\sqrt{-1})\)

by substituting \(\tilde{I}_{p}=K e^{i \omega t}\) (K unknown) and its derivatives into (8), and then take the real part \(I_{p}\) of \(\tilde{I}_{p}\), showing that \(I_{p}\) agrees with (2), (4). Hint. Use the Euler formula \(e^{i \omega t}=\cos \omega t+i \sin \omega t\) of [(11) in Sec. 2.2 with \(\omega t\) instead of t1. Note that \(E_{0} \omega \cos \omega t\) in (1) is the real part of \(E_{0} \omega e^{i \omega t}\) in (8). Use \(i^{2}=-1\).

(b) The complex impedance Z is defined by

\(Z=R+i S=R+i\left(\omega L-\frac{1}{\omega C}\right)\)

Show that K obtained in (a) can be written as

\(K=\frac{E_{0}}{i Z}\)

Note that the real part of Z is R, the imaginary part is the reactance S, and the absolute value is the impedance \(|Z|=\sqrt{R^{2}+S^{2}}\) as defined before. Sec Fig. 68.

(c) Find the steady-state solution of the ODE \(I^{\prime \prime}+2 I^{\prime}+3 I=20 \cos t\), first by the real method and then by the complex method, and compare. (Show the details of your work.)

(d) Apply the complex method to an RLC-circuit of your choice.

Text Transcription:

L tilde I’’ + R tilde I’ + 1/C tilde = E_0 omega e^i omega t (i=sqrt -1)

Tilde I_p = Ke^i omega t

I_p

tilde I_p

e^i omega t = cos omega t + i sin omega t

omega t

E_0 omega cos omega t

E_0 omega e^i omega t

i^2=-1

Z=R+iS=R+i(omega L-1/omega C)

K=E_0/iZ

|Z|=sqrt R^2+S^2

I’’+2I’+3I=20 cos t