Mathematics notes.pdf

BSCS ,BSIT, BSMATH MATHEMATICS notes of Author "DENNIS G ZILL" EDITION 9 mathematics solved notes of chapter 1 and 2 all questions step-by step solutions are here

1. CHAPTER No |:
"INTO Io

Derroran uno”
“Dirrrentiat Eounmon”
ven;- UD-E)

e
An equation. conkining the

fone or more unknown 5
—atable ie 424609) Jundten with

eu to one or mare independnt
Variable „is called diflenentiai aquabin”
Example: __

Hrn
Ars Of D-E:

dy CDE LOrdinany eifjorentrad Equation)
PDE _(Pankial diffencatiod Eapation)
+ ODE:
An. equalion containing the_
ordin ay —dosiuibive. ef “ore ot.
mac depende uosiable with
iebpeni_ To IE one at mas

2. eh salle” FE
Ge ts dendkd Y ES
Examples ——
- a = Sy =e de
— 3h ye?

de a dy = Gxt À
EE :

> PDE =
_&

An apokion “contig Tre
———pottic_denivalive of one or —
® — More... dependent able with
à - neapecl Jo. -One_ar_nare_tnde=

3. dependent

aes ary Dependent= 4
Srégente-dt Sodepgent = 1
ee dependent
ra day _ Dependent. &
cpt - Gndependert= L
Independent

2 Nomrrons: __
_—)___esmtz Notation:
du_, dy 434 any
dx De” hé

Lab Prime. AE A

Pry

U des of DE. eon — ES
highest dleiuatve In. de pion?

Scanned with CamScannc

4. cae 2

i ”
EN upon poor of detudte.

= Cxomple:

aa ds rad, + dl

de de A

ll Orden q DE = io
Cana Of Differetial Equations >

cn “heaps 4, 15 the_poxer
heichest ne ta the
__equion’

“Boris >
“aha ue a

5. 4 pet ip

N fit orden of osdeñinasy differentia?
Audion. in Some ime weiter. tn

ent foun.
Musa + NU 4-0

= ma fam. g- Siena
ooh torn of DE
an A rn ep

— - ee yy
Eee din =
ic fran aa

Ses orme} asın. ap Oe.

- ar A A nn
el fee) en

sh — Zac.

en rn SENS
ces) -

ae

6. 1 Example if
| cay Netw fo 4D

fue ode.

A ar =d>

aa ses — fist cn PE
1 = 5 ae

DE,
RN 204 sa $ Normal {om ef

tu Agpönls
À El

RE Dirrerenteran Enupmon
BR Frias =) = E
An den ODE ‘i he
eng LA = fogs =)

o said to Rincon y E — E
a e a

7. Noter

Two pwpeties of linear ODE:
4- We- dependant — _vætoble"
_desivaMwes. y Y. ef are 4 the fut
_ degree. Men _ ¡Ls pawen.g cach arm,
imodiiog_ 18 4.

en Ayres Ln. 4 ata 2
Abe. idapendent.. vastable

y cnd.all ils

pasat Suntec we

u

“ODE. sen steph

ae a ee eee
— een, ite

8. + Non-\ineas D'E
Novibezz, pin.

amp °
cane
(dy e O
per ne! À
Ly + =O

a

Exersize QD 5

_ Dékesraine whhdhes. the esting
ts Mneor_ac non-linear. _

r Ay) Uy do Aa cos | : =
dincan..—2md_ecyelen___ a

050) y! = à
Pe Wird_arden =

9. “ey tae

Non- rn q ¿re ovclen
2 a d+ u = Coslr+u)

nn- nasi + Rod ardor.

_—_.Non-linean.,-arcler à.
Y Detesrnine_whethen. ‘he. “equation |
— Is fist orden differential _ a
— tn linen intote dependent viable -.

Eg ratching fust_ DE
Y ld ndo in.

10. Implieik:

amt
ANS At? seprile
Tara ependert..and dependent ;
aaa. — —
a+ ná =
Espias : .
= N Sun 10 a segle 5
td erro Er. en of
uch orden vg a E

Ga y EIERN “Aled one

11. aed

premio e fly 4 ddl ton.

a E pond oy alutionge
Flyin hs

ulm) is.

ER Fall of DE
constant (er sf
y» ere La

Tas, gi Gi,
1 Postical ss

fukin ct. cares rare a
Id puasosic. eppedisn is lid — --
: :

i pee ee

O dilution of Det that exist
shy one oki ts called
Sogelen _lulion.

== eukon nix sa
ld et VE:

12. 7 es ap 28 te i Ma
queer =

AAA
PTE ET

13. de 204-24 % DE

at

WM soy 24 ==
(ven ES
3 4-8 ee

4-2-3

Diferenticile. wert OC
¿et = une

14. A E —+—
Pros

y andar de

sa yy ———

E

pre eee
= _Be“cosla|

|.

st Bee —
4% si Ne
= ra =
O = Bett —

E
, a year AM
ji —
(pe _——_— _Q ES

15. om. Ln LEE X + LON K ) case [secta
seca ane x]

a len]? Bee rms car]

ge Pure xx 0 os, fans asec ]
Ts atan] - cose | sue
gu stafosent y au. au)

BE sido] art han Simel 4
(een Franc) Cos

ss

—stondaf pa Maia ys

= so fat —— E
e So [reci ceca] esoo

sentloma

—— x + 4 |
———— gg mu intern —

Hor stos ln per ten) —_
pa 0 00

16. ‚Amer. cosas 3h

Leon = tome —

3 Ney aide unción ya ee
cree re given — ande =
ae ordex- a equslion..

ey FE a RUE
Genie e428

3 sea teilen is f2,a)_.

loves scenic dicción. ac EN
debition. th 2,

17. % wen! CHER WAR

.._ Dimain of funchon ty La)
gros as is a D ©

Aluen yanaalırı —
_Dilfenenbate wert.

18. a Dia pats The parto
AA)
A tata CRETE
ars ss DE —= —
Co
A N= Stansn EA cc
E Treg @Q _
= 25+ÿ 2 _

RSC = 25+ a
Dösecrän = 25 + 254ad
AS se Sn 225+ en aa 1)
secó 2549 Sec, _ LE ne
Se Sn = DS cee En

19. eneattéle wirt In —
a = er) 2 2x

Eu of = Ct

By ay eosu q ll si NEA

Ray 4) cos its DE

2 - 7
cos y 2:

20. N.
(1 sion) "+
_Diffenenttiele sk te
Ae Rees Bs)
EN use] es

toon
2 int Sen à ae

O —
Msi)

= _É Los 0 = |
; RE xa. Ei

: me
: — sx =
Ñ 5 SE ;

Se

21. D .
a N) Um) ‘im 2-1) st

are tx DA). is DE
at :
As Mer. ft

to AA i IE ean Ov mi eee
pa in em, un ee

AK En DMA
E ee er
RAN) MA = u) Du
WIR) RU) if

Venu goed

22. Ley ap ae
___dilfenectiött — Ute
BERN sa 2 na ES

23. ak

ep. ice) ae

er

pu)

24. y

25. een AU
N. er) pid PIN
ra

O: i |

da 28 Tata

12 de Agres UN

: pr ae)

er Y aa
e : wie Ady -0

26. == Diffesentie ae

= dy = STE Tat pe u
= =

RER |

- put in D a = 7
pr -
ee e
a ed
—— nn

29. Á fol" pda sent E
\ ©

+ Wand

Ser ee ng

a

= | — Late re a = al

31. wah OME

dy.
Not ae

MA on

a et

a OS

A.

3) se os
=~ Suns, . a — —

mx

— Ya

33. ra ay +2y'=0 yen?
7 wy! ae —G
jen ee
Ye met FY = ml UN

Mm ae

put in À
ft em pet FE me :

xhtml) ion ot Dame +6

x ve

=a

AO) Aue) E
—__ ec. condani goth. —_—————

BN
c= PR sl a
21 E

ares na

34. = llentiol ef the
cen
ach

_ (osm-.- Foy ay
ae

va, ¿0 7

RE ao AC
___ Asa Ihese one cole rattial condition

A

35. —— mae no 1: ;
| Crtven DE fs q
Given seluion
Aa condition
yo =3
yure2

gece —Q)

aC ms

ia 1
wich ts sepia, teal

__ alu. Problem...

36. Exqrcixe (1-2)
is one penamebric. pa, y

_ DEA fiset ocdenis y'=4-4* —
Find. tolulion À Tsk ade IVP is
TT onda of Da, queen andihien. a
AM RO A NIE P

sin = | ous
9 —

40. yeeros” Uwo ponamtrie
and anden. pny =e ™

Initial < conditions —

42. EE

Equation (Y) “cecomesy
pa ay
Sesa „ce no

es

pur in@ ä

5 eo) ne.

= -5.- 5 0-08

3 _gluenolubion ron =i Be E
———— pub. Valse a re E

—Y e
haa

44. E q dx

ave __gneprife_
aes \ny = Wan sine
ng Maat,

nee anlog bes

46. su © Arey

Se da
O a.

EI
ET er
Ortega,

Host À ER SE
2 —— ir Tages
5 144 ar MOTS)
3

47. m:

“ad

ae AA
e) Ae) A
AN wa —— Fi
MP

48. E a SENTE
U My y

A

A ION

| A GE
EUR ES re
E
: Ne ae
‘À > AN
MA resorts |
a M2 dy = 4-1 dx
= 1. MPAA 2 A =
= hi pao
L en 4-4) 4)
a WL ata
E
fur

PAT.

EEE a
aan FEB PT eng si

PY at bal se ee

en] za + |
FPT NA
AA AOS, a
AE “a. = Lua
Kram
RER ll

E EE = 5
cea ARE

hacen rer

NE —

+ YA

49. El =
ras A eC
Le

51. le O

4 eed

52. is combinen at Corser) hut tt
à Wr
-__dertualie ig ur elornen!

Fu DRE US
4

53. JA

Aspa = (Dect) de
= ARMA
2

54. Arena
Oh ee sig OC. =0.-

a nO > sin’. Bel a

CEE à Oo.
RE. | 5 +

ie
Sl 2D

1

ae se te,
ae

La ae >

55. arcas izan,
A ¿gc oe
ET Si _

ÿ

Ons

56. gy so lodo!
a stand td J

y dy = sind —

wee. +608 #0
EN

Le A E -
rast ——

creerlo CES

——akie®_volues 1

57. Le? a
4 oa

ax

_ a
E

59. else

posit toto to Jada e

‘Scanned with CamScamner

60. O ol
A

a 4 a
roo

ele ————

ie |

E bc eee PRE RES

0.1 0-1

61. IESO

ar

Ast A-sine)

AA

= Sas = Le ax

A SO An

sat

cos

Axe

63. wi free Pare

urur vat EN nat
inal tait

Dalits 2b MU à ne
Nya Law) ental int

fetal bla
olaa oz ee

Ass

64. Liar Equations

a pud-ouler affine equa
of - tha Pam
pe a. at y ga
xs sta tbe - cena equalon in the
__-Nosictole Ya ”
_ Sonderel form
— = a

ae
BEET go ct leg Qu +
A E

ee Lo FRS
—— 072. Menard sy

pale . cata ón.
ue aR Asus pion |
— —— YN e | EN

— Aleps_Io 1 Ale ni se
— fe linea ga

65. we equalisn with Dire edf,
Jara Mo. lern, a . A

myo Hd Se ye palio. pa by ca
er ‘
eco in he pa ~

66. SUR ta Sandero. foo gu alin,
SS interim Lado.

A EMO.
ax R

68. sat

“ga sanded JA

DA
ALAN xe

ET aN

3 SE

“en

69. o. ae

A
A
sean wi Cm

Mn rn

A

70. nn o ne
j Ne e Sanden Zoro at ei”

Ma
"td y à Lou

dia y u
GA —

5 ea SR

EC M Et

71. A pee —-

Nas a =

De

ad + y =

aA fut ul
Degen
y y) AY

72. degli | bs
Aa. Lt ash E

BT
—Y AM A BER E

73. g- ANA Ay re ns
La IN_Iandend

—fosen.s

Pod bua) fun. ae"

we |

factors
$

a Er D
ARAN wetter) ¿>= Le
i an

74. cc ca

a doo

que = qu] |

= AA “én “Jandel am,

— 0——

75. prune ten o

ze rear —— er
mie Te

Sy a sed as

76. gneycavo expebon
sec) Lane rl

o Looe. +

sec pen

y

y = Siow cose Ss wee

Be Naha GAN} nd —

er = rane
+ (KD y ~
a. ary) A \
coe PU Peay as
nal a —

e”
A tat

E A
MdB: (Anode =
e dl Ped wee th

(ust ah + oer Aad yee MP
E Fat

77. oS ee PON
mpeg joué pie

78. a À

gin puso a -
Mat ai NE car 2
EA
El Ben e ee y
N “ie handen ya] 6

io ai A aha a 5

Teens

aa ay ASS e 5

Afuralal = Sins
am! Ir

= if

79. # a À réecb

EC

stand to osas me ”
A ESOO > 099

Ba ee

N: see —— ya o.

2 PR} EN =

a DO Fine.

| = ira

amas urea) ccf

80. Fa ent wen
ax
on grade em

a

rave ET en gen a
antago peter + - pus

_ Multi
say Bundend..eqpälion. —
sith

= pe Jadat

=
xe dis ad See en nook
aed ef)

nat nes à) rely —

83. a see ie, qu Den
ag AQ.

‘Scanned with CunScanner

84. ET
AE E E = TR u

MR Side £
= a" $ F

sr \ ee 7 2
— Ago did Brent
Inieqgating, Laci . u
MX di + Ri = € et
At E, Lo

Sea

85. 3 | ad A 08

== “On dlendend u fama j

dr Pisa Te

— _ — A
E a — and Jam ih
ES

x en =

__ ———
de oe

nat ty
a ”

ae

wi Conc

86. ERES — (1)

a be E A

He WALT

87. Im
4 {oss +

em +e es

Myattng- th ——

oo

ago que pe um .

Bar ie et zn

1 —— =).

E: a Ce
De AA

ds yy Al rl =

a ae
put in WAR)

E MO oO -Q+C

= ADC

1 Cour ae É A

Mas a a ES 4

EE i

88. sea ad PET =
A A
2 AN quo is

A = = ;

saadead | Be

_
Multi ins

TER né + qna dt

fey} dons

Ton bh

89. inc ivi intial
ales Sis amen

D ago re au
nal ass
ni

Scanned with Cam

92. dal nda y
ey U) becomes
4e Qa Lee

93. ees ren elos wk

94. e* ue

ut
yore fe Peta]

sha trs RL 50 tha
——_ aka in follveuing manner,

AA

— | ER ep) RE

96. 1 She vine integral, fundion
is defined as

sin) © ee

we the irlegral is defined to be
_ A ok. 0
.. ..Euple tha. afin a Ciel vale
MN pscblem. ss
et wa Daly 2 ose qu
in Ama A
Ws es. stas.

97. A arepa af ran — =
U zo fSialn)- an]

y 7 Ss ö E
ap er A u Zuge

98. tT LQUATIONS
fnaktoni-

A vot Mong de ¥NOGgldy 5 on
nic Atffenenticd toa lara
A xt. camserpmadenee do he Airacnlios

A O ai Ps
Laden. DER [sects
Milas ri cha

did do be ened Retin of Me open
— un cd: am oes 7

a.

E SR

Listas)

À 3 theo fee Yun a

= ae

BET

MOE LOE OE SE AT CR ie L

<a ES gy o a
are pd a
E 3

Le,

99. brertise em ~~
Of given Affenenteen equation.

Seinen wave tt em
O a

; (Rx tigo
ot Wed

+ Ne 39471.
à

Es ~~ = EE AK. ed DE
er)

Sa Bf in que hg e
a ES

— ad y E”

a ee

101. y > A a,

oy Rn “Hy e a m
My = vr Nu dea’
dy own A
pen fatal 1 mol exe.

a Appropetale. ‘Wegnating. factor

a
__ a MN pa CC
N

mM

Byres) = poo | cd vu)
kin)

Cw) 2e hy)
- | —— QE
y Ta |
TAS _

—— TT a Ne

SS pr acme ae

te ad

103. ori “E = G-mody

AX y
Ñ Pon my «o a

© Nandonet Jown rs
Min de No dyno E,
Lampara Mm.
ge —

104. ces
sh,

cost
Fa -Q

105. ay (aba do + LL 4 an) dy -
ne Aare

——Miisadds 3 Noa de ye

May Ne. Lan
\ 4
ET
Hague ae +
Not exect = ts
becuse Hk So
am + aN

EN EN

106. MA pad
{pean {erm no sm E

pone nag ia —
E

A i. en

(lary ASA A AT Sur A LR
CEE nes
A 3

Scanned

107. A = Dek
y a = TR -\ + Hea
\ = Method
Pd ve de
a. Jam a
_ Mbooyldn Nfld o
(Axo 4 + ow Jda- mit 20

Ta tue Mr Ne x

ela a, va 7
Nue : [an ;
| Spe pee = -
a ia

108. holve- que

ES pa A
Sendra Jr
rcs

us =

109. Ar ir
: Lata
Pause lL,
pur Tn ea 0)
; a (Ay 228 Sik à (oye 4t=I)dy20
= Sandee. 4 e0=2

Má -_Leron free. froma kd Mes
aka May aC =

111. NOAA

QA

Find value oh MA 86 that gun ven
neck? ns _——

a AK IR dut y +Lonty dy so

39m Sanden farnas ©

= Mayda tun ray
Malta ido

url _
on Es

am „am
ST _s* ENTE
ilies Def Aloud o Un
De dû — —
2 e E
— (Guy? -tesy)de + ira bans sing dy
METEO EN à

EE
SM Spas, —— — Ne 28 AN
SM. Ving? siny = troy sing -

112. \

\ Ye \

\

A quen DD 7
N 7
Ae qu DE by dee 4
e 4
a E
Ma on ls 4

tit A Ee

113. me Leagsinn x
Aman +
4 Cero

ey sin rx
Pep com|da + (2, ca
3 + ac at ‘chs a

)

A gc, acu

TM
aos
y SN AL ft LOS, N 2)
= y cos

N ye Sox Be IN 24,
ee + As species N= dots 2
pe JO gan

4 Fe ea ai ee E

ee

ostia

114. TT topos pa
: Perce La.
apena foren ——

gars Ne)

= Oy a a
DR
am. gy ON
SN Ea B_
am LAN
a ar

My; on

N a

[arpa y

mA ao a ES
LRO >

\

GNA

Worded 9 ä
Ra __ wie

115. Sa ea —
Vino «is mule,

nas Lis free He Si: e

2)

=

AE =C

3 Guy das (Oy sot) dy =9
mies ___Nande pi ==
ER Mata.

SS eae

es AD

116. |

|

E id A ee Oe Hee |

117. $ NO 64 + Edn

M 10- sure” dy so
ihn D 7 we
EZ Bo |
à AE
u pi O ei (ds
6. Nol ower
; Cas Homer - er dad,
SE er
u en a |
-6-0 2-6 vn = ae

u SA 3
—% CEE

= e
mt -Stondend. eafiation. ath Lk
_ RL

enix, oo

118. penne

ae we 26 y
3

posé) =
D

e

\
= me -2y1 Es

119. © Slandend gotea Yeo
wir AAN wld à # e.
Nam EOS

121. ee + de se IC
20 | + de
=
— e | | =
a 4 e |
= E |
= >.
+4 :
OC N |

ve Ae o
pS
a =2
“A
=
A 29

122. ee

=
nee

ny ade hing Sep —

123. NA Ae OW OA

xX ¥

E A EE
ES Ns *W,) A

ser A ea
po md] sa)

TR Nhe_equalion. ts Porn

rampa |
eb gala apa Ay). wi un

ee E DER
Y A,

mr CEE]

ud == ee

ax E AA

124. z BL LT Wa Gat |
+L | And 4 av
eo seen =o

ot eo) ae oe ee

125. e

E

Mandy + We ans
x

w Y

126. AA CA

NN
MA SIAM)

y LS ts)

127. rc (95 )
Questions 1-10

3 EE Faye

A

MAN y) di,
= a =\+
—

|

AC)

E putin ed
e 5 =
IT an a ——

ar ei
Ir m :

= M
|

\ |
u

wi. -t

Ne nr

Raimi = — =
&

128. yd
laa TT

u u — Ml
3 y dy 50

A ate =
Ada Y) 0
A RA E

pal gane
ne da equ) _

sandy. =
dk Vx-2x

NES CURE LS | LE
de 1-27

129. ‘aw Li»
M à (tees
CE v2

Yo Wevevell |
ax vr

EIN

130. i qu v= 3,
Blu = A x
Y 3

y

yo 7
Yaly-x) 2) e+ |
m-Y |

yla x ACEON

131. eee ETS ar en

Rurale a ot

132. aM = (nani

133. ” toto! ol wi y ne
Agnus AA Fall E
put. Hs

€

Zu + E Im ya wy? | Ho

alas A) aba eas xn = 2

= ig + dE

134. = Ye 43

___ ge -given-jatial a value problem;

acl)? (ur Xela) a | Dy?
Cuad cus

135. | Lyleg x
grating,

137. ee
Qi
| AY sy Poe a

u. Mm >

| Med
[ger

7

| a Bernalli's. equation 15,

Dee N Rady à fy"

139. _DERNOUN. Kauntzon

er = À + My Qo gr
Where : : m+0,1
pt atte

141. A
y $ 1er)

7 ep ny

ARE re:

N j ; Es - ~ Vera's
EN | en Da E.

142. RS sin. =
st E pao Lun)

Ne nie" PET

1 3
| = ui ec]

143. où faq M

at r 1“

ß | ET eee es
—

145. A
j his Ds | ea
, MUSAS der

Bio) se
x

bin
E

Az) wot Hy y mn

v

147. GAY
a eee

à AM]

a wey

MR
_ 4 DETENTE ant eles
> a o
AA
bs ae ee

RS

148. dy dz ——
TO

de lo dar
EN

7 de taz el
e

de ea
Fogar

wi a E
TED
lan E
tana - y 3

2 = Pal
secta

149. : fovacde A alada».

ta
- y Fe Laa. fax
|

Sa) > 449) 4 €
+ Sa

— BODO) ARA 4e

IAN
ina) = pala (a 5
— 7 acc

sin WHY) = ZOE WAC
UI. _
B à - ANS
+ CAME eer
dehyde ppg adios
ea a slit

150. ANA =o tC a —

34 Last) ya, —

e aren 22

— - ee

ee oe …

e
A ea a Be
A dee eae ee ee
ax =
— a Paola cia
dz „ Arc
a RI ud
WA c
Ax cos —

151. Re Vaina

a

i A= COs
3
|

(1¥ 05x) (146052) vai

Lose
ÀA-cos 2

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> | SEAN =

de - ax