A summary was made about theme 2 before attending weekly tutorial sessions. These are cripnotes that cover the fundamental theory to better understand the application of each lecture unit and how they follow each other in the theme.
Theorom 41
The tVP ha d unig ue solution yia) on an interual
the function'Sdh n-1, 0o
ard g(a) dre contnyouJ cn tno inteyvd
AND dnla) 70 for dll xtL
exmple
y69 (-1) y dll g1e continuous
ard del(x) tO
tz) d,(x) g.()
IP gtr) 0homogeh eous eq ugtlon
Theorem 41.2.-
1f yy
1f y y , . n _are olutjons af 0f on on inte1 va
1, dhy inea cenm binatíoh of soluticnJ
y C1y Czy :Cnyh wi)l dlso be
q solutich on .
depenoenf t Wronskíqh(
Lineayly
(inpe pende hf if wIonnk iah( t0)
* * *
n On skidn
wit,tr,tnff 13
f
P
Theoiem 4.15
It ty yt, ynIJ a funddmentdl set of bf on 1 then
d doned golutloh Of bE S d
hinea combjndtien ot y,yt Yn
* *
, Sinco
ccs-Sin
tah9 Jec
geherd Jolytjeh 1VP ccrdtjonJ given
Denve gene dl S
get value of consjanis.
T o not Viol dfe Theorem 411
4o be
congiahts 9, (), q, (a) dola),9ix) nood
NOT 0 Udl fo zero
Ohalysed
get qeherql olution
Venfy fundamenta| Set ofJolutjonJ.
denvo functich g veh to 2n ovder olerivdtive : veny Jod
tqike qenerd| Solutich dro got agtn), a,(7), d4la)g)
See tha they'e CcntinuouS every where on the inlervdl
perferm wionsk iah with functiens. fo seet inegily
indep /dep Cirdep t )
i t in Indep tnen d coloirq +o def/Theorem 4. 1.3
functien form a fundqmentdl jet of solutians.
10 find gehea solutich
m
m yyy eg uo/ to
Homo equdtjon Jef
y c Cconjt aht)
fuctcrse
inspection
long divsjon
+greuplng
9Uaddti eq udtich
20
) 1o find specific gjolut jen
gef geheo Jo.
derive generd Joluficn Jupp jn ordy to get
Valu of constqnts.
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