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Plan for today 5.2, 5.5 A comment on RLC circuits Learning goals:

Be able to solve a linear 1st order system for which the corresponding matrix has characteristic 1.equation with complex roots or repeated roots using the eigenvalue method.

Announcements/Reminders

Solutions to Quiz 4 posted on Gradescope 1.Read the textbook! 2.

Last time

X s A A52 9

GO.info so9Y

4 2 4

1 Solve char edu to find eigenudet Az 1 O 1 0 IL

2 For 4 0 found eigenvector

so ee is

1

For t I i find eigenvectorNon zero A E L i GA IF o

Eti i H L

Elementary row operations 1 interchange rows2multiplying row by non zeroseal

Iadd a row to another

I I I

I to4a

I I I O

D L L

II I I 0

I L LU Vz Vz O V 3 vz

4Vz 43 0 Vz 24so

f3 is an eigenvector for anyv2

Recall if Xis an eigenvaluecet

sin

IE.it oi

gemforany2

cettFor t 1 exercise find eigenvectorthen general soly

x C f J t caetf cc et

ET

p signfrom K Iliu inders bee eigenvaluesare distinct not obvious thatdistinct eigenu give nlin indersols but the

Cakofcomplexeigeurate

exµ

A 14 II

Eigenvalues of ItDet E t 4 G t t 16 0

13 if 163 X Ii 4

i

conj pair ofroots

1 3412 Find eigenvector

A 3 4 i I y 0

Iii II Is4 i v 4 vz Vz Iv

Gui 4iy u i ySame

so 2nd win doesn't add infoe fit lifeeH4ittfi is a soin

I 1

Reed looking for 2 liu indep sols

Option Li find eigenvector for D 3 4i

it will have complexentriese3tab a b cplx

Jeu Solin 4 1 Cz 2

E TCplx const

Option 2 Observe

if thx real entries

the AfRe check thedetails

If solves I It thenthe 1M also do

Take The lies of e 4itfiEt if E 4tiin i

Euler's formula

est fcos C4Hi sin C4H

cos cat t isin 4 4

it II tie

Real pt y e f ITIt

im.pt y e't L IIFound 2 SolsCheck that they are lin indep

W y gee3tC sinC4tHe3tcoscqe3tcosC4t1 est sinC4H

e to liu indepSo gen sota

yc e't fine IIIIt A

real coast

Summary for cpk conj eigenuFind eigenvalueFind eigenvector car to one of themFind cplx valued Solh e.g e HfiProduce 2 bin indep sols by takingreal imaginary pts

RLC circuitsMechanical systems mass spring w damping

external forcemx ex t kx fCHT T I Tmas damp Sfroingf force

To 1or c

same

Inductor Lhphgsd em

WU meaningTTresistorvoffffe Rohm

charge in capacitorQct satisfies c

L Q RQ E Q Ect

It R I IQ ECtldt z DI

dt Currey

Can predict behavior of mass spring bybeh of equivalent Rlc circuit