Section 2.2 Homogeneous with Constant coefficients SOLDEi
Pto Review i General and Fundamental Solutions
Part L Guessing Fundamental Solutions
Wednesday HCC SOLDE Theorem
Review Fundamental Solutions Them
Part 2 The Complex Solutions Case
qq.ggly
yo g comp q
o
Review General and Fu Solutions
Them If Y 72 with J f C Yz are SoCs of
LCI o LCT o
with LCY Y t a Ctl y t aocts Tthen every Sol Y of
aycan be written as
Jct c YCTI t Cz Y Ct L
where G Cz Cours Es
Remarks i Y Y above are fundamental solutions
y in L is the general solution
Guessing Fundamental Solutions
Example Find all sols of
Y t 5 Y t 6 I o
LCT
Sec We guess i L Y Co const or
rt21 Y f or 3 Y e or
4 Y sirs t or 5 Y coscat
1 LC.co co t 5 Co t 6 Co
6 Co
co 0 6 Co 0 120 0 1 Nor nonzeroconst c's
Failed solution
2 Lct th 5 th 6znh Cn c th 2 t 5 n th t 6 th
Find ns.t 2 th o all E
n Cn l t 2t 5h t t 6 th o all C
Check i No solution for info FaoCeo
3 L Cert rtg
rt6
rt
r2 ett g r cart 6 cartert r2 5 r 6
L Cert o r't 5 rt 6 O
Y ett is so l of Y t 5 Y 67 0
I Ir EIR is so2 of t6o
ascharacteristic equation
per r't 5 rt 6q
characteristic polynomial
r 511 54172 5211 YE I2
so ye 2u ageocs.fun.gl
LCT
So all socs of Ley o are
ctD8
HI SIDE Three
Det The differential of
Loy Y t a Y t do I O a uo consts
has the character y
per r2 t d r t do Oe
Thr mi Given an HCC SOL DE
Y t a Y t do I 0 1
Let r be the roots of
th t d r t do 0 2
A If of toe real or complex
Jae eun so a are
g Ct e g Ct er t
B tf tf E To
the fund Sols of CL are
Ct ebt yet zrot
Proofs A See example above i
y er L Cert ert art aoert
Cort r2 a r t do
O r2 t d r t do 2
Since rt are sols of 2 YI are SoCs of
B y ert o Lce't ert r2 ta r too
o r2tacrto
But now a do are such that
to c's the only root of
to t a Ro t do 0 Cbc
roosarootithatmeans
to r diI2
Therefore 1042 400 0 1which means froe.az L s 2roto O BL
to c's the only root
so Y er solves Loy o
We Look for a second solution of Loy 3 0 as
LCT o is Find a Ef fort
HIPS i since 7 is Sol of Lottothe ef for weight be
simpler than the cof for YT
Y ta Ys t do Yz o s
Vy t a cry t do Y o1 1Product Rule vY
t v1Y tVy
1 y t v y
V nd t V Y v Y try
V Y t 2 V Y t V Y 1So i
v Y t 2V Y t Vy t a V'Y.tv Y taoCrY so
Y t v 23 t d Y t V Y ta Y tacit O
Now Y e't y r Corot y rozerot
r
v t V 2 ta t V ro ta.ro too e O
v tV 2rotatVCro2ta rotao
but 0 0
to is the ro is a rootonly root
So V J the eq for V c's
tianya Ea.geThe sols are i
VCt cto c d consts
So i
Tact Vct Y Ct y CtS at ebt 1
The constants C d are arbitraryBut i If we choose D 0 that's bad
because
L cent funnfantentaSoCs
So we choose CEO OL L Then
wts te
BE
Ex ample iFond all possiblemovements of this
1 keg systemµ
4otb
Vy
SI Newton's Law i
Yy byY o Y
s i Y t6Ylt9Y
know i yetrt
ert 6 ert 9 Cert o
r2 t ar t 9 ert o
r2 t 6 r t 9 0
re 6 I 1 62407So2
611 222522322
rt r 3 1So the Ihrm above says all sols are
yct c e ttstE Me
Example i solve the IVP
Y t 3 Y 4 Y O J
Y Lo 2 Y co 3
Sofi Wo look for fund Sols of the form
Jct Ert
Cert 3 ert g ert o
r2 t 3 r 4 O
r 311 32 401412 311 9162 2
of 3252 rt I
Irso
4 12 5 1 fund SoCs
then lycts ctettc.EE general Soc
Initial Conditions Yo L Y co 3
Soi2 Yco C e t e e
12 4 2 1 a
3 Y'co C e t C4 C c
13 4421 b
e c 2 C and b
3 C 4 2 Ct
Ct 8 t
4938 5 Ct 3 Ct
C 2
2kg
E
Ic I
so yct et_ze I
Top Related