Engineering
Mathematics
partial differential equations
4
clarification
2)
Lagrange 's
a) Type
b) Type
4
-
41
Heat
5)
Wave
I
-2
Type
31 char pit
method
's
-3
→
Multipliers
method
equation
equation
-
a
-o---f__f__p___fI-o
gÉ-g _
-
paihal differential equations
flu y )
2-
-
where
u
y
,
①
which
w r t
-
-
/
is
i.
e
dependent
is
4Y
.
y
r is
constant
s=%÷y t=¥yr
In
equation
which
2)
pole
which
is
H,
said
is
partial
first order
only
1)
A
:
f- (
F- ✗
-
%y=¥y /
jhz
an
a
on
w.at
z
contains
partial
.
First-order pde_
contains
①
constant
is
r=
A pole
derivatives
it
y
q=%y
DEI :
Difft
,
variables &
depending
is
u
-
%ñ¥u
p=%u
independent
are
variable
Difft
①
-
,
to
first order pole
derivatives
it
.
of the form
y;
2
; p
hyp tug
,
g) =
E-
a
0
-1g
ÑÑ%-ntYEy=ñi
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__lT-_↳÷A
Faculty of Mathematical and Physical Sciences
-o---f__f__p___fI-o
gÉ-g _
-
clarification of first order pde
Linear pde_
be
linear
of
P g.
i. e
F-
×
if it
linear
is
+
yip
ñyq
+
linear
functions
in
of
i.
Ex :
re
e
1)
-
only
the
coefficients
.
play ) p
pole
said to semi linear
the co efficient of
is
-
pole
psg
if
it
are
QCN.yjq-RCN.Y.at
pole
the
sing
=
'
'
-12
said
is
co
-
to
be quaei linear if
be functions
efficient may
variables
play
it is
of
linear
dependent
.
,
2) P
11
n'zp
+
21
Ñy
p
Every
linear
F-
lineorpde
:
order
independent
+
ntyq
+
but all
i. e
a
.
Iuaeilinear pde
p sq
+
z
p and q , and
and
y only
hyp
A first
xyz
pde#osI
A first order
"
and
z
to
"
=
rip + y 'q=
Eibar
and
,
said
QCN.yjq-RH.ylz-scx.gl
play )P
2)
in
p g.
is
:
1)
is
in
functions of nay
are
z
.
partial differential equation
A
:
"
+
ICKY
y2zq
+
=
,
semi
converse
=
12174,2)
Ñy'z
"
but the
q
my
Key )q=
pole
2)
linear pole
is
not true
→
euaeilinear pole
.
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__l0-_↳÷A
Faculty of Mathematical and Physical Sciences
____-f-II-p___fI--
ff-o-ct-mfytkgngpde.is
-
niyp
1)
rig
+
2)
Pt q
3)
My zp
4)
ÑY'p
5)
UP
+
n' y ' q
+
ygL=
'
dependent
F-
✗ :
pole
pole
is
-
variable
'
-
A pole
derivatives
F-
✗
-
I
:
said
its
Tzu
l
+
Not
pde_
pole if
terms of pole
D
21
0¥
Y÷y= nyz
are
of
not
0¥
In
-
e-
non
÷%
( wave
variable
s÷y
+
÷
,
homogeneous
-
order
same
,
-
+
.
all its
contains
partial
either
.
equation)
.
:
said to be
is
linear pole
derivatives
=
*
a
pole
pole
partial
,
He
pole
homogeneous
be
to
linear
Linear
→
→
pde
oeeaei linear
→
y
22
dependent
on
:
-
xyz +
=
or
homogeneous
depend
x
of same order and all the
2)
Non
semi
→
rly 'zq=
+
Linear
→
+z
I
=
Homogeneous
A
derivatives
My
=
.
In
-
-
and
if all the partial
contains a term that does 't
pole
+2
=
my
ng
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__lT-_↳÷A←
Faculty of Mathematical and Physical Sciences
____--E-f__p-Tf---→og--___
-
Lagrange 's
i. e
where
method
Equation
p
R
I ✗
.
first orderpde_
to solve
of the form
pp
functions of
are
ruled Lagrange
Qq=R
+
y
a
,
ez
.
wg
write
the
auxiliary equation
's
d÷=d÷=d÷
2)
choose
Either
d÷= d%
d÷=d÷
dig
3)
solve those
The
two
like
lolutions
41
U
Either
wing
u
solution
is
absent
is
absent
is
n
=
c
,
4
for the
0
arbitrary
or
or
absent
is
talk
or
or
YI
=
,
,
function
f- 2)
=
out
cancel
cancel
to
one
out
find
Cz
given pde
Uff
cancel
methods
well known
film g)
( ci.cz/--
2
Either y
,
general
where
DIE
=
from
equations
two
any
is
0
.
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-og-r-ng-ghog-IE.A
T
____-f-Ih_p_-→fI-fÉ-g _
-
Example
sold
:
:
The
-
solve
D
-
µ÷)p+uzq=y
given equation
-
of the form
is
PP + Qq=R
where
The
P
YIµ
:
D=
az
&
,
12=5
Lagrange 's auxiliary equations
are
d÷= dig d÷
=
¥-4
choose
d÷z=dy÷
=
,
2nd term
its
①
in
-
①
,
dy÷µ=d÷z
÷Ie=d÷
n' die
Integrate
n÷
y
=
Eg
=
-
1st
dy
both sides
on
¥ ¥
choose
-
and
+
c
=
e
3rd
we
,
get
,
-
,
terms
②
in
①
.
¥1m
se
du
-
=
YI
zdz
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-oco-fi.AM
-___--__f__p_-→fÉ-o
ff--___
-
integrate
U÷
÷
for
obtain
by
choose
general
solution
is
¥ Y÷ Eg E)
(
.
-
-
and
=
p
of the pole
1- any q
+
=
tanz
auxiliary
d,÷ny= dt÷z
2nd terms
in
equations
-
①
,
Cota
cotn.dn-coty.dz
on
both
ln ( Sina )
=
-
.
Cost
Sinn
sides
.
lnfsiny )
bn
given
are
①
d÷nn=d¥ny
Integrate
0
,
Lagrange 's
d÷ann=
1st
③
have the
we
the solution
The
-
( ci.cz/-- 0
tana
-
Cz
c-
=
pole
the given
u
sign :
Is
-
+
,
Cl
②
2-2
=
and ③
②
From
both sides
on
we
+
get
bnc ,
(
iny.ci/sin#y-.e
lm Sinn
=
,
-
②
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-ofng-ghd-fi.AM
-___-foIh_p_-→_É-o
ff--___
-
Choon
2nd and 3rd terms
d÷ny=
of (
e
,
,
a
-
+
luck)
③
-
solution for the
given pde
general
the
eat
.
sina.cz
=
";÷z
.
lnlsinz )
=
sing
sides
both
on
lnlsiny )
is
get
cotzdz
=
integrate
③
we
,
d÷nz
wtydy
From ② and
①
in
-0
&C÷÷y si;÷d=o
.
③
solve
sign :
-
The
yap
The
-
seyq
xlz
=
given eqh
is
2g )
of the form pp
Lagrange 's auniliarg
dig Hay
=
Choon
-
1st and
2nd
%
ditty
-
-
-
ndn=
%Tz
=
terms
of
in
①
is
+
age R
given by
①
-
ay ,
,
ydy
integration both sides
II Eg %
we
,
get
-
=
-
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-ofng-ghog-IF.A
T
____--E-f__p_-→fÉ-o
ff-g-e-ai-yt.ci
-
n'+
choose
and
y
dttey
Cz
%Tz
dy
dz
I
-
I. F.
el tody
=
tg
.
e.
=
ay )
-
ydz
=
Zy
2¥
+
Zy
+
z
=
-
=
%y
get
( ←g→)
-
=
+
②
d÷y
=
ay )
Ty
we
,
=
-
-
,
term
Ly dy
-
c
=
3rd
and
-
-
lost 81
2
2-③
¥⑨+Py
y
=
The solution for the differential equation ③
y
z
.
=/
yz
y
y
'
=
Y
From
2.
dy
-
y
② and
a
fa.I.edu
+c
is
+ er
+ en
'
z
=
Cz
=
④
-
we
have
the
,
for the given pole
of laity
④
'
general
solution
as
'
ya y 1=0
-
,
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-oco-fi.AM
-o---foII-p___fI--
-
ff-o-type-2-C.pl
> +
aq
)
R
:
Rules Lagrange auxiliary
working
tepi
write
-
equations
the
d÷=%-= ¥
step
a
Find
-
a
solution
wing
d÷= dog /
dip
dz /
,
dog dz /
=
fi
ray
step -3
the first
Use
Itu
-
general
µ( fi
Exampty :
sofa :
-
The
1)
Thr
it is
If
it
free from
free from
is
If
it
is
free from
fz
a
P -13g
given pole
is
of ④
④
=
f,
52
+
=
dY- did
choose 1st and
day
=
2nd
pole
of the form
-
is
)
given by
52-itanly.sn ,
term
3N
PP+Qq=R
dz
=
is
§ ( Fz )
tan ly
Lagrange 's auxiliary equation
=
from
Cz
=
fz )=0
solve
@
,
solution for the
given
,
z
-
equation
two
remaining
Thr
a
If
solution to find second solution
say
step
c
=
following equations
of the
one
①
,
did
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ngr-LE-FF.AE
-___-f-Ih_p_-→fÉ-o
ff--___
-
3dm
integrate
=
3k
and
1st
both sides
-
-1C
y
y=e
,
②
,
3rd terms
equation
in
①
,
dz
dYd
-
on
3N
choose
dy
-
=
g-
"
z
+
fancy
-3k
)
s.dz?--+an(-e ,
=
3N
Y
,
Integrate
fdn
=
N
=
n
=
n
The
general
/
=
fdn
both sides
on
sz
÷)
Is
¥
(
+
tante
52 +
,
)d
-
y
,
N
-
3N
=
,
-
C
,
"
)
tante , )
(
52 +
fancy
solution of the
given
of ( 3k
-
,
g- z
bn
C
-
? Fante )
+
Cz
tglnfs-z-tanfy.sn ) )
-
-
Y
,
5
bn
-
Isdn /
52 +
-3N
pole
+
))
Cz
=
Cz
-
③
is
tardy -3N )
)
=
0
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
-___-foIh_p_-→fÉ-o
ff--___
-
Solve
②
sold :
-
xzlf-nylp-yzli-nylq.it
given pole
The
The
a-
p
of the form
is
auxiliary
Lagrange 's
dy
=
✗
choose
2122 + ny )
2nd terms
and
1st
aoe
given by
dz
-
=
Ia
yz(z2+ny ,
①
,
dy
du
=
NY
PP+oq=R
equations
du
-
T2
)
yzfz4nyy
-
÷=%both
integration
lulu )
.
=
truly
-
but my /
my
choose
sides
e
=
②
,
terms
du
(22+9)
z
is du
=
/
d÷
=
=
get
we
,
d÷.
=
( z2+ny ,
is du
lucci )
,
dn
U2
+
luce )
=
1st and 3rd
NZ
/
we
=
n
=
+
24
+
"
"
-
-
my
(23+92) dz
,
"
-
,
(22+4) dz
integration both sides
4- 2¥
¥ ¥
u
c
z
-
e.
2C ,
get
+
"
z
2nyz
+
C-
=
ez
'
/
-
G
:
my
③
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__Lg-÷É&
Faculty of Mathematical and Physical Sciences
-___-faf-_p_-→fI-ff--___
-
From
,
of ( ny
①
③
solve
sin :
my
xyp
+
F
yfq
"
"
-
z
of the form
'
Lagrange auxiliary
The
=
given by
-
zny -2W
2nd terms
and
is
dz
d÷y= dytt
1st
Pp + Qq=R
equation
's
choose
'
2x
-
>
-
-
X
is
given by
any 221--0
-
zay
to
is
"
of Cu at any 4
=
=
given pole
the
-
n
,
solution of the
given pole
general
the
② &③
①
,
d÷y,= dfte
d÷=
integrate
dig
lnlnl
=
lulu )
n
choose
2nd
-
and
I
dy
dy
dy
-
1g
dy
sides
both
on
truly /
=
+
,
lncc , )
but ye , )
ye ,
-
-
c
-
,
②
3rd terms
,
dz
=
any -2W
dz
=
"÷!
-
÷
•
dz
=
zloty ) -21kg /
~
dz
-
zc ,
-24
'
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
-___-faf-_p-TfI---
ff-o-inhgs.at
-
both
on
J dy
-
.
say
=
Y
Y
Y
From
-
=
ze
,
25
-
)%,%→
:-,
÷
÷
In
lnlzc -292 )
hrlzc
,
%
-
sci )
-2¥
ttn solution
,
a)
+
,
but
② and ③
5)
"
I
ez
/
ez
=
,
41¥
H-
,
dz
,
-
sides
,
Y
-
solve
solve
1- but
py
+
/
for
e-
the
-
1g
③
given pole
is
given by
-4--2%111=0
qn=nyzYÑ-yY
NZP-iyzq-a.my
Faculty of Mathematical and Physical Sciences
=
G=
.
.
@ Ramaiah University of Applied Sciences
-___--__f__p_-→fÉ-o
ff--___
-
TYP
Let
p
Q
,
,
,
R
,
functions of
be
,
my
42
.
Then
well
known principle of
algebra each function
by
will be equal Itu
's
equation
aniliary
Lagrange
a
,
)
pidu-Q.dy-R.dz//fP.P-QiQtR.R
term
(
If
P
,
which
P -19 I
,
-1-12,12=0
method
This
Us
lay 2)
cz
=
,
The
G.
s
give u
to find
Cale
p
,
lay -4--9
another
,
-
o
.
integral
Q
,
-
4
,
,
R
are
,
called
.
for
.
to
this
In
.
pidn-a.dy-R.dz
the
repeated
is
multipliers
i.
integrated
be
can
then
,
in
the
given pole
flu .ua/--
is
,
0
solve
①
zcu-iyjp-zcn-yjq-oi-yt-p-rso-ni.LI
Lagrange auxiliary equations
's
du
ZCn+y )
choose
P dn
,
+
Piz CU
d.
→
Pix
Qi
-
y Ri
-
as
z
n2+yÉ
22
,
-
,
①
multipliers
zedu
-10,21N g) + R /
-
n'+ye
-
dyi-R.dz
y)
=
zcu y )
.
,
given by
da
dy
=
are
✗
=
-
ydy
-
zdz
/ UTY ) yzlu y ) -2C n'ayy
-
-
du
-
y
dy
-
zdz
H+#H+-ÉH
ndx
integrate
-
ydy
-
U
?
y
zdz
both
on
÷ YI
=)
-
-
'
-
o
=
sides
we
,
get
3- %
=
22
c
,
-
②
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__hE-_↳÷A
Faculty of Mathematical and Physical Sciences
____-foII_p___fI-ff--___
-
choose
y
N
,
-
,
second
as
z
out
ydn
ydntndy-z.dz
yzlnty )
tuzla y )
-
-
zlx
ydntudy
dlyn )
integrate
From
②
&③
zdz
-
s
.
ndy
-
zdz
-
0
=o
sides
z÷=cz
-
+
H+#+¥Hz Ézyf
=
zdz
-
the G.
,
'+yy=
both
on
yn
of multipliers
for
,
-
the
③
given pole
is
cfcnkyi.ae?yn-I)--o
②
solve
lmz-ynytp-lnz-fg-IE-ly-Y-znns.ly
Lagrange auniliaryegf
m÷y nndtf-z.dz
's
:
is
①
=
ly
choose
t.m.sn
as
first
ut
-
mu
multipliers
of
.
ddu-imdy-indzdlmz-nyj-mfnn-lzj-ncly.mn
edu-imdy-ir.dz
=
,
edu
integrate
IN
choose
+
+
mdytndz
my
a. y ,z
both
on
-1 nz
as
=
C
__
0
tmz
-
lrfytmnn mllztny nnfn
-
-
( t.mn
are
constants
)
sides
,
second
-
set
②
of
multipliers
udu-iydy-zdzIYMII.ly?IyY#ez)+z(ey.mn5-umz-
nyy+yyn yeasty -7m
-
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng_-LT-_↳÷A←
Faculty of Mathematical and Physical Sciences
____-faf-_p___fI-ff--___
-
udntydytzdz
=)
integrate
Ig
both
on
Ig
+
sides
÷
+
af+ye+z2=
from
Solve
sotn
ly
my
+
+ nz
2) Pt ( z
-
y
choose
Ndn
-
for
.
the
given pole
is
n'+ y -124=0
'
,
-
n
)q=
dy
=
(Y
-
N
y
-
Lagrange 's auxiliary equations
:
du
✗
③
ez
,
& ( IN
③
kg
=
the G. s
and ③
②
o
=
2)
-
y
a.
ydy
+
+
ylz
z
z
d÷y
=
n
multipliers
as
z
,
-
zdz
-1
ndntydy
=
-
a)
+
-
both sides
on
zdz
HUH -14/2-44+24 -2¢
zcn y ,
sedntydytzd
integrate
→
-3=0
,
÷+Y÷+¥=÷
214 y
=)
choose
1,1
,
'
-12
dn+dy+dz_
Y
-
2+2
-
u
1- N
-
C
-
,
①
multipliers
as
I
?
fdntdytdz )
=
o
y
du
+
integrate
✗ +
dytdz
on
y
-12
=
both
=
ez
o
sides
-
②
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-L
T-fi.AT
-o----E-f__p___fI-gÉ-g _
-
From
the G. s
and ②
①
for the
.
,
of Luty
classification
at
-
→
,
y
a +
+
2)
pole
given
=
is
o
of _É
ofpde.it
comet integral
com¥
s
An
or
expression of the
pole f- Cu
the
,
y
Z
,
,
P E)
,
=
o
complete integral
called
type
where
-
,
of Cu
Y
,
,
2
,
and b
a
b)
a.
are
0
=
is
arbitrary
the
solution of
constants
is
.
pairs integrate
2)
particular integral of the pdefcn.y.z.AE/--0 is obtained
and b in
complete integral
by giving particular values to
A
a
General integral
3)
An
expression of
functions
said
is
the
type
be
to
Singular incl
4)
gal
or
or
.
solution
of Lu
general
,
v1
=
o
where
solution of
the
ofln
,
y ,z a. b)
,
= o
be the
is
pole
of
Now
( if
it
the
)
enist
arbitrary
pdefcn.y.z.P.gl
-_
o
.
of the
pole
flu.Y.z.P.si:O
.
y=a
envelope
is
of
complete integral
alto solution
of pale flu
.
y ,z, b. g) so
,
Y=b
:
If
Itu
vase
.
Remand
that
and
singular solution
complete integral
If the envelope of complete integral
called
exist then
it
singular integral
Let
u
envelope
pole has
of
no
complete integral
singular volution
does 't exist then
we
say
.
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng_-LT-_↳÷A←
Faculty of Mathematical and Physical Sciences
-oG_--E-f__p___fÉ-o
ff-o-char-pitsme.tk
-
first
Itn
coneider
Let
f- ( n
,
y
,
first order pde_
solving non-h
of
order
non
linear pole
-
2,1391=0
working Rate
steps
steps
Write
.
given pole
the
¥+÷÷=÷y÷e¥
¥n=fn
Let
dp
find
and
stupa
SOI :
-
two
P sq
¥g
4
-
terms
-
for
dy
=
,
proper
The given
Let
pole
da
in
1K
in
②
fq
auxiliary
egn ②
+
qdy
integral
writer
be
can
Pdu
-
and
hence
integrate
.
complete
obtain the
D
:
fp
dy
qfg
-
complete integral
find
Examp↳_
Pfp
-
.
substitute
.
to
-
-
,
=
egeeali
or
q
¥
,
dz
fytqfz
p
¥z=fz
.
-
¥
•
-
=
select
.
-
¥
.
fy
-
=
de
=
fu + pfz
steps
Fy
,
①
.
-7¥ -1¥
dz
=
→
0
auxiliary equation
eharpit 's
write down
.
fcn.y.z.P.az/--
as
of
z=pn+qy+p4I①
Pntory -11045-2--0
as
fcu.y.zp.ge/=pu+qytP~-g2-z-@Difjt-f
partially
w.tt
u
.
,
fn =P
char
,
pit 's
dp
fu + pfz
fy=q
,
fz=
auxiliary
d
,
=
equation
is
dp
,
P E
,
+21'
fq=Y -12g
,
given
die
=
fytqfz
Z
,
a
dz
"
=
fp
-1
y
=
-
pfp
-
qfg
dq
by
=
-
fp
dz
¥2
t-X-J-q-q-J-p.n-ispj-qly-s.si
From
dp -0
1st and
the
4
dg
2nd terms
we
③
%÷p
)
=
%+É
,
have
,
:O
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
-___-faf-_p___fÉ-o
fÉ-g _
-
integrate
p
a
-
dz
WKT
integrate
which
②
-
µdp
a
-
fdg
-
b)
by
+
obtain
we
,
fla b)
,
required complete integral
the
pole
Given pde
-
sides
+
an
=
bdy
+
both
on
solve
sold :
adn
-
the
is
get
pdutqdy
.
2
we
,
q :b
g
-
dz
sides
both
on
q=3P ?
-
①
be writers
can
.
q -31,2=0
as
flx.y.zp.az/=q-3pt-@Diff-tf
partially
Let
w.r.tn
fu 0
char
pit 's
fy=O
=
dp
,
fz
,
auxiliary
,
is
dp=°
integrate
Pfp
-
by
µ
-
¥2
fp
du
=
=
•
l
=
-
dz
cop
q
③
dy
-
1
,
both sides
on
-
given
qfg
-
=
term
1st
the
fq
=
-
d÷
=
P&g
2.
du
dz
fytqfz
d÷
4.
-67.4
=
futpfz
From
=
equation
de
=
fp
:O
,
,
p=a
From
①
D=
WKT
we
,
have
g-
sat
q= sat
a
dz
pdn
=
dz= adu
+
-13A
integrate
2
=
which
are
is
qdy
on
+
the
-
dy
both sides
salty
.
+ b
required complete
integral
.
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-ghf-fi.A
T
-___-faf-_pTfI--fÉ-g
_
-
3) obtain complete integral of
sod :
Given equation can be
writers
-
Corridor
Differentiate
fu
The
=P
fntpfz
DP
Choo
a
-8
1st two teams
¥
pfp spa
-
lnlq )
=
bnfp )
substitute
=
qy
-
pz=o
.
,
y p
given
-
by
-
-
pi
-d÷q
,
-
-d÷p
,
,
both sides
tncp )
peg
+
%
=
on
pre
.
-¥= %
=
-
fg
is
dz
,
integrate
&
,
d÷=→d÷ gig
=
p
,
f- p=
,
f÷gfz=
=
w.r.t.se y ,z,
auxiliary equation
char pit 's
d"
partially
f- 2=0
,
as
fln.Y.z.kz/=PNtqy-P2
f
fy=q
,
pole px+qy=pq
the
:
,
lnlal
+
lnlqa )
p : qa
p=qa
the
in
pole
given
pn+qy=pq
qaktqy gia
=
arty
-
-
ga
arty
q=
a
i.
p=qa
=
an +
y
&
anti
g-
-
a
WKT
dz
dz
=
=
adz
pdntqdy
( an + g) du
=
alan
=
adz =
g)
(an g)
+
( ant)dy
du
+
Can + g)
dy
( adu dy )
+
Canty ) d Canty )
integrate
az
+
+
=
on
both sides
H÷
+
,
b
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-gho-fi.A
T
____--__f__p_-→fÉ-o
-
ff-o-wzkngruh-tofin-dc.in
#
step '
complete integral wing
Find
.
4 IN
steps
steps
Difft
.
of
solution
Mold
,
Y
a
,
b)
,
w
2%
4
=
0
a
and
r
-
integral
the
of the
get
we
,
¥b=0
b
¥a=o
wing
¥
&
-
find
to
o
-
singular
,
solution of pde which
and is
constants
arbitrary functions
particular values of a and b in
singular
from
Find
and b
a
solution
the
is
arbitrary
obtained
by any
4)
method say
.
0
=
t
-
prev
.
The
:
2
,
partially
Eliminate
:
solution
,
is
free
not
complete
.
pole
22N
and hence obtain
integral
complete
-
poi
2qny+pq=0
-
singular
solution / if exist )
.
Conti der
flu y
p.se/=2zn-Pn--2gny+pq
Solin ?
Differentiate f- partially w.r.t.se y ,z peg
-
,
,
2
,
,
,
fu
=
char
22
-
pit 's
dp
2Pac
From
pfp
team
2N
=
given
d"
qfg
fp=
,
by
=
-
y
Ñ+q
-
=
-
-2mg + p
doe
dz
pl n'+ El
fq=
fq
=
-
,
dy
-
,
-
El
-
any-117
dy
=
-
fake,
-
1- yay xp)
have
we
,
0
=
gg-
-
222
both sides
on
-
given pde
in
a
-
pl
Is
,
a
pn 2-
a
=
'
-
ie
ganef
)
=
2k
+
"
¥
Faculty of Mathematical and Physical Sciences
Pa
zany
( ay
=
0
-
-
a- a
,
-
-294+294=0
integrate
substitute
is
fz
,
=
dq
+2p/m
dq
equation
-
=
2nd
-29N
=
dz
fytqfz
sqy
fy
=
dp
-
,
"
=
-2p/m
284
aumliary
d
fu + pfz
22
-
-
22N
2)
2
@ Ramaiah University of Applied Sciences
-___--__f__p_-→fÉ-o
ff--___
-
substitute
P&g
dz
-
ady
-
ur
2k
dn
=
ay
✗
h
by
-
a-
z
dz
1- y
-
integrate
ady
z
both sides
on
-
aka
-
ay
ay)
partially
①
y
=
-
which
soju
:
bulb )
blah a)
-
-
-
w.at
a
-
①
but
ba
w.at
b
the
pole
1-
a
ñy=ylu- 64
rely
solution
required singular
complete
aka
=
✗
-
partially
①
o
z
,
the
is
pifft
b
z=
of the
1-
-
b
-
that
implies
Find
/ n' a)
bn
=
.
.
y
5)
du
→
required complete integral
is
-
①
2k
ay
lnfz
Difft
n
-
=
2-
which
ady
+
-
ady
-
qdy
Calf 2) du
2N
=
a-
dz
+
2
A- ✗
dz
Pdn
-
lay -21 du
2k
=
da
in
and hence obtain
integral
212 + Pntyql
=yp~
.
solution / if exist )
singular
.
Conti der
ffu.y.3-p.se/--2zi-2pn-2yg-ypDiff-tf
partially
w.r.t.n.y.rs t.se
,
fu
-
217
,
char
fy=
pit 's
dp
2g -15
,
g- p
-
2
f- p
,
-
equation
2N
is
=
DE
sg
-
phony
2YP
-
pfp
-
given
qfg
plan
-
gyp )
by
=
-
y
du
-
syst
dy
-
,
dz
-
f- g- 2y
-
,
=
=
"
-
d"
dz
"
fytqfz
fu + pfz
-
aumhary
d
=
dp
fz
fq
sypj
dy
-
ay
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-ofng-ghd-fi.AM
-o-_--__f__p___fÉ-o
ff--___
-
choose
1st and
÷
dtp
last
-
29dg
=
integrate
both
sides
2hr14
I
on
(p)
but
-
-
-
)
P
P
p
=)
p
212-1
.
dz
✗
h
=
my
Eye
ydz
ydz
gy
+
p
q
die
-1
/
y
=
22
-
ay÷
.
-
+
-
dy
-
¥
dy
=
ya
is
,
we
-
the
-
+
2a÷ ) dy
-
dy
dy
qdy
+
-
+ a
ad (
zdy
( ty
aiytdy
-
du
-
Kyu dy )
Eg )
get
aIY÷
+
atg
+
b
①
required complete integral
singular soln
Difft ① partially
For
The
2a÷ )
-
-22
,
.
sa
¥
→
=
÷
=
safe
-
Fy / ÷
agdn
-
,
dz =p du
in
q
integration
which
pde
,
dcyzl
on
/ g-
Ty 1¥
g-
substitute
a
¥
=
luca I
a
=
=
=
luca 1
=
given
a÷
qy
luca )
+
)
'
y
.
in
-
y
-
p
substitute
,
2hr / y )
+
but
2
,
Fay
-
,
bn
term
.
.
given pole
w.r.it
has
.
no
b
o
=
I
singular
solution
.
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-ghh-fi.A
T
-___-faf-_p___fÉ-o
ff-o-sezdord-ri.de
-
clauif_Én
linear
The
seed order pde_
of
second order
pole
AÑ÷~+B°÷ugy+c
Where
A
B
,
C
,
,
D, E
SF
,
hyperbolic
elliptic
clauify
E
1)
3
Sok :-p
+
F-
Fy
,
+
is
FU=0
.
said to be
o
B2_4Ac=o
if
1374A
eco
-
pole's ;
the
2)
Rewriting Itu
33%2 Fy
>
-
if
°÷nu= Hegg
DY.se
constants
B' AAC
following
thin
+
real
are
if
parabolic
✓
8-
Far
3)
÷
=
given pde
Y÷ -1%2=0
as
-
observe that
A :3
B ? AAC
The
i.
2)
-0
-
=
13--0,4<=0
,
0
given pole
Rewriting
pole 's
the
%÷z 0¥
=
-
parabolic
is
as
0
,
observe
-
that
A-
-
132-4 Ac
:
i.
3)
1
=
The
13=0 S
,
=
-4 (1) C- 1)
0¥
1,13=0
132-4 Ac
+
,
=
,
-
=
given pde
Given
A- =
C
470
hyperbolic
is
0¥
I
-
☐
,
&
→
Given
pole
is
elliptic
.
c- I
0 -4
-
(1) (1)
=
-4 Co
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__h
T-f÷ÉÑ
Faculty of Mathematical and Physical Sciences
____-faf-_p___fÉ-o
ff-o-coneider-wpdeRr-ss-tt-fcn.Y.z.PE/--
-
①
0
where
r
-53
,
R
S
.
T
-
s=0÷sy t=°¥u
-
-
defined
partial
1)
Hyperbolic
a
5at
5-
Elliptic
3)
at
RT >
ART
1)
Nagy
12=1
②
said
is
domain
in
to be
if
D
0
Cay )
domain
in
☐
if
0
Cny )
domain
in
D
if
0
?
,
T
5--0
,
↳
0¥
=
(
equation )
Tricorn
N
,
RT
0
=
pole
given
-1-0
n
0
=
5- 4.12T
Th
<
①
ltupde
Aunt
s
=
point
a
5- ART
Efg
( a. g)
point
a
D.
equation
point
4
only powering partial
nosy
the domain
in
at
parabolic
2)
of
differential
The
S
,
functions
are
derivatives
,
4G ) ( N )
-
-
-
is
4N
hyperbolic
if
elliptic
parabolic
✗
it
if
<
or
a > 0
,
a- o
-
.
( n'
yygte-ny-xyig-py-gn-2lnl-y-ls.ly
my
,
-
-
observe that
coneider
The
SIKRI
given pole
R=ny
.
s=
-
( n'-74
T
-
-
-
my
.tn?y4t-4lny1l-ny1--(x?yy4-an'y-
is
{
parabolic
if
y
elk
a-
-
-0
-
hyperbolic
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-gho-fi.A
T
-___--__f__p_-→fÉ-o
ff-o-supe-rposifnprinupIIfui.uu.us
-
.
.
Uk
-
differential equation
U
where
a .cz
-
Cia
-
solutions
are
,
e
=
U, +
,
Cruz-1
-
+
-
-
constants
are
a
linear
Itu
thin
of
CKUK ,
alto
is
a
,
partial
linear
homogeneous
combination
solution of
pde
Itn
Money vibrating string ( wa_veeguatw
:
it
place
to
ends
string andalongfasten
1k
n
lengthand
,
n=o
the
it
home
problem
is
that
string time
at
any
The model
One
-
dimensional
equation
wave
since
the
boundary
release
We
Assume
it
t >0
find
vibrations
deflection
its
vibrating string
%÷=
ie
c-
gy
fastened
motion of
initial
=
④
f- in)
coneisls
where
c
the ends
string
velocity
Equation
,
"
①
Fla )
implies
G
"
will
n
Ctl
=
T
,
-
-
linton
g=
=L
dimensional
we
,
have the
•
depend
initial
its
on
③
of
the
¥
duuihi
.
pole
①
.
④
pifft
,
partially
④
'
w.r.tt
Ft Flu ) G'Itt
Difft partially w.r.t.it
ice
Gltg FIN G' It )
Glt )
F IN )
=
↳ ✗
x=o
be the solution
Gltl
w.r.tn
,
÷
=
2¥/t=o= got )
w.r.t.se
0¥
any point
one
①
at
&
partially
¥n= F'In )
of the
,
the
uol.tl = FIN
partially
at
Ulu.tl
.
a
its
[
✗ →
the
-
ifft
:
0°
.
Itu
,
and
Difft
ucn.tl
"
www.oeucr.t ,=o
,
☐
T
.
to
,
←
UCH 01
-
t=o
is
is
deflection
:
the
determine
string
conditions
Furthermore
sold
then distort
to
of
equation
wave
the
it
.
,
and
we
.
time
to vibrate
at
,
The
=L
a
stretch
,
at
L
and
string
and
allow
of
anis
-
=
'
'
=
,
that
c
'
F' Tn ) Glt)
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
-oG---E-f__p_-→fÉ-o
ff--___
-
separating
variables
the
"
G Ctl
"
In ,
F¥
=
c-Get ,
observe that
but
LHS
two
then
constant
home
"
(n )
"
+
put
n=o ,
=
function of
RHS is
&
when
it
is
'
n'
equal
only
to
-
G÷÷µ
and
o
=
②
UCO .tl
×
-
,
& ④
we
G
"
-
-
.
✗
,
t.tl -1 c-✗ Gl
-11=0-50
have
,
Glt I
Foil
=
=
and
Fen /
✗
Using equations
uol.tl
only
'
t
possible only
is
,
→
=
g.
F- 1117
this
-
.
"I ÷+
F
function of
is
equal
are
obtain
we
,
1--107911-7=0
/
-
-
UCO +1=0
.
.
1--107=0
put
a =L
,
UIL,tl=
F- (
F-
i.
"
( n)
Gct
FCL )
L
)
-_
UCL.tl _-
1--121=0
&
→
-
calf
solve this to
find
Fen )
⑥
7=0
Equation
implies that
⑥
F' ' (a)
on
0
0
1--101=0
+ ✗ Fln )=0
I
1=0
integrating
we
,
n=0
n
f- Cn )
get
=
Can
Flo /
,
=)
put
=L
F' Cal
S
0
=
5- ( n )
put
,
+
ez
=
Cz=
FIL ) =
,
c=
☐
/
-
=
C
,
Cnn -112
I 1--107=0
0
C
,
L :O
I
FLL )=0
4=0
i.ec
i.
i.
FCN )
From
=
=
,
cz=
0
0
equation
④
uol.tl
"=o→
This
is
pole ①
trivial solution for
which
is
of
no
the
interest
.
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__Lg-÷É&
Faculty of Mathematical and Physical Sciences
____-faf-_p_-→fI-ff--___
-
cased
✗
-22<0
=
Equation
F-
The
"
( n)
A E
that
2- FIN )=0
-
1--101=0
m2-22--0
-
.
i.
Flat
put
implies
⑥
2--0
'
=
[(
o
MEI
"
e
e
)=
put
=L
a
+ ez
,
C
=)
=
,
FILK
,
e-
-1C ,
e
,
1--121=0
4
-
e
=
I
0
"
-
Cz
Flat
e
e-
+
=D
④
uol.tl
C
4
✗
FThe
"
A E
is
.
'
__
i.
have
we
This
→
o
trivial solution of the pole ①
is
is
of
no
inleresent
.
1--101=0
ME
Flo )=
,
L
-
a-
Colfax )
c
FIL )
,
1--121=0
4
.
Flake ,
n=o
n=
0
=
by
given
=)
put
,
1=0
that
implies
-1×2--0
M
put
⑥
( n ) -124--1217=0
-
1--121=0
22>0
=
Equation
-
,
which
Casey
<
<
:
Eo
<
e-
1
0
equation
the
From
1--101--0
:
-1 Cz
"
czc
i.
-
Cz
cze
-
real roots
✗ k
ed + c- e- 4=0
,
→
-12
m=
,
+
m=±ix→
ezsinctn )
costs 't casino}
C
,
=
/
sink 4=0
=)
LL
F- I
⑦
FCN )
implies
=
-
NIT
,
✗
=n
that
czsin
(
I
1--101=0
0
I
Casino 4=0
=
roots
⑦
-
=o
Complex
.
n=
✗
-
1--14=0
i
Take
1,2 , 3
=
✗
.
'
Cito
-
for non-trivial sobn
.
.
=ñ¥→ Eigenvalues
,
NII )
⑧
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
____--__f__p_-→fÉ-o
ff--___
-
G
the
equation
solve
Now
A.
given by
is
E.
'
;
Get /
.
CLEO
-1
m
=)
czcoslctnt )
=
ME
+
substitute @ and @
ucn.tl
find
⑤ to
t.tl -1 c-✗ Gl -11=0--1
"
Cr sin
-
-
Gltl ,
G'
dI
'
Lt ) -1cL
m=
↳ since ✗ t )
④
in
( MII )
↳
-
Gltl
-1 ice →
Un
-
n
'
we
-
un
-
n
have
we
n
-
-
put
.
i. 2.3
.
-
-
-
-
un
/
( NII )
-
}
-
uol.tl
An : GG
Bn
-
Cola
-
,
cost
n)tBnSinCcn )
'
i
n
t=o
in
-
-
ten)
-_
.
flu )
)
we
find An
An
=
4
Bn
④
.
2¥/t=o= got )
&
① and
to
uol.io/--
flat
is
Ula 01
sinfn
I
conditions ③
Ulan
i.e
put
An
=
initial
the
use
:
sin
,
P
W
UCn.tl
solutions
1,2
)
t
+
get different
By superposition principle
MT
=
I
cos(cn f) casino
Ancoscctj+Bnsin(cn)
-
✗
,
MI
For each
complex roots
/
⑨
-
0
-_
sin
/ n¥
,
)
="-szsa--→-*
which
is
half
range
Fourier eerie
.
Zfotfcn1sinCnII1dse1 ODif _t@w.rt 4ueeY_1t.j-gcn1.p
An
It
-
-
f- Anc
=
since
"
)
+
cost - t)
Bnc
n=l
)
sin /
n)
p
4-+1+-5-8 " /
Bnc
=
n
Faculty of Mathematical and Physical Sciences
-
-
sin
( NII
)
l
@ Ramaiah University of Applied Sciences
.
____--__f__p_-→_É-o
ff--___
-
which
is
BnIÉ
half
¥ ! tgcnsincn
±
÷
Bn=
solution for
Ducati
where
-
the
An
=
n
[
gents in (
,
The
cost
An
-
is
deflection of
n¥)dn-i@
⑦
t
'
I die
)
is
+
by
given
Bn since
I f. 4-in siren
÷
,
which
pole
,
) sinfn )
l
Bn=
↳
Fourier eerie
range
the
the
[
gents in
required
string
at
non
-
I die
NII ) du
trivial solution
any point
on
<
L
4 at
any
that
time
represents
t> o
-
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__hI-_÷÷A←
Faculty of Mathematical and Physical Sciences
-___-faf-_p___fI-fÉ-g _
-
Heat Equation
Consider
thin
a
lengthandL
of
rod
temperature
tent throughout
zero
for all time
limp
temp Ulu .tl
at
.
at
.
k8¥n
Uco t )
Ul
n
,
01
f- In )
②
022cL
Assume
-
t
.
find
to
L
}-@
and
at
uol.tl
> 0
temperature
the
time
any
egf
2¥
Difft
partially
④
Ñ÷
"
✗ (
=
Equation
X)
w.at
home
then
two
are
constant
①
Difft
partially
④
w.r.tt
✗ cnltltl
=
,
"
✗ (n )
-
Fi
-
,
function of
is
equal
this
only
'
t
-
RHS is
s
possible only
is
,
function of
when
it
is
'
n'
equal
only
to
.
'
-
☒+ ,
×"✗÷g
"
pole
✗ (a) Tilt )
=
1- It I
✗
any
,
÷÷+
but
at
n
-
that
the variables
LHS
>a
Tct )
"
observe that
[
Yf
K ✗ (a) TITI
separate
g
the solution of the
w.r.t.ae
implies
①
yu=o
.
'
teen partially
y
of the rod uln.tl
✗ (a) Ttt )
=
0
④
-
Difft
t> 0
be
✗ (a) Tct )
=
Us
t >0
.
③
.
is
obtain
Then
.
①
.
04 LL
point
SOI ?
=
whose ends held
0 < n <
,
initial
an
.
UIL t 1=0
,
problem
The
¥
,
0
=
,
time t
any
=
-1>0
with
=
(N )
-
¥÷
+ ✗ ✗ Cn )
=
and
✗ 101=0
=
-
✗
,
1-1+1
1¥ ,
and
✗
From ②
-
4
0
④
4
T
'
=
Its
+
-
✗
✗
KT It 1=0
have
we
,
✗ ( L)
=
0
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-L
T-fi.AT
____-faf-_pTfI--ff--___
-
"
✗ Cn )
F-
✗
=
✗ IN /
integrating
N=0
=L
✗ (
,
L
=
put
✗ (
n
"
)
✗(
.
(n )
-
-
o)
e
put
✗ 14=0
cz=0
④
,
we
get
of
solution
the
pole
which of
@
no
interest
.
that
d- ✗ CNI
=
,
✗
0
__
mIñ=o
m=±a→ real
mkI
roots
eM+cze→n
,
I
C , + cz=o
Cp
=)
N=L
:
0
=
given by
is
N=0
4
0
equation
implies
⑤
✗
:
)
✗ 101=0
-22<0
Equation
-
92=0
-_
,
in
the trivial
is
A. E.
/I
0
=
)
✗ in )=o
Ulu.tl
=
0
✗ CU / = 0
substitute
✗
KTlt1=0⑥
✗
-112
C, N
✗ ( O ) =cz=
,
i.
cased
+
get
we
,
C
which
⑤
0
=
→ Cz
✗
1- Itt
,
✗ ( Nt
put
-
that
implies
⑤
"
put
-
0
equation
on
'
O
-
4×14=0
✗ 101=0
calf
✗ Cal
1- ✗
14=9
=
-
^
✗ 101=0
-
.
Cz
"
+
e
cz e-
✗
2=0
I
'
-
.
✗ ( 4=0
czeft-cze-dk-oe~C-et.lt
-
e-
=)
i.
put
Cz
✗ Cut
in
✗ In )=o
=D
<
4--0
4--0
&
0
=
equation
uol.tl
__
0
→
④
we
,
get
Trivial solution for the pole @
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__hE-f÷i&
Faculty of Mathematical and Physical Sciences
-o---faf-_p_-→fI-ff--___
-
calf
✗ =L
Equation
"
> 0
✗
The
A.
i.
put
✗ Cn )
2=0
✗
,
"
log
C
put
n =L
)
+
+
(2)
LL
.
✗ cut =
n
-
⑥
i. e
'
11-
'
)
→
find
to
Iti
=
-
-
-
=
-
we
,
k
→
complex
roots
✗ 101=0
-
ez -1-0
✗
14=0
for
nontrivial cob
)
-
=L
→
=
Eigenvalues
.
⑦
.
'
-
TLE )
~
get
lnltct ) )
bnltltl )
tnl
=
-
1T¥
-
Kitt
bug
)
=
3
T¥
Titi
=
-
↳
e-
=
-
-
+
lncg
ka 't
kit
K - 4-
e- KII
/
't
-
"
I
Eigenfunction
Tltl
Ka
,
integrating
+ is
Ct ) -1K ✗ 1-1-4=0
t÷÷
on
:
-
czsincn
equation
)
I
Take
✗
-
solve
m=
0
=
o
=
i. 2,3
-
=n¥
✗
:
(
,
NIT
=
( LL )
ez Sin
=
sink 21=0
=)
a-
Sinful
Cz
I casino ?
,
-
0
=
,
✗
,
c
=
ME
m2+< 2=0
cos ( 2n
c ,
=
.
-122 ✗ Into
Cal
given by
is
E-
that
implies
⑤
Faculty of Mathematical and Physical Sciences
D=
2¥
'
⑧
@ Ramaiah University of Applied Sciences
-___-faf-_p_-→fI-fÉ-g _
-
put
egn ⑦
UCN.tl
For
then
⑧
&
obtain
we
,
czsin (
=
each
'
'
solutions
e-
ucn.tl
¥1T
"
solution
.
Eun
-
n
=
't
n
-
,
2
,
3-
solution
-
to
(
/n )
-
differentiate
An
-
Crs
of their solutions
rum
[
1
-
Un
is
•
Ula .tl
=
sin
By superposition principle
a
"
different
have
take
An
=
1¥
NII ) ↳ e-
we
n
Un
④
in
An e-
Ñt
sin (
is
)
abo
NII )
l
-
n
-
I
(
uol.co/--fCxt):.uCn.o1--fCn1--EAnsinlnI -)n-put
t
This
-
-
:
.
half
is
An
The
and
o
-
2-
non
-
we
egI
③
e
.
1
range
[
flat
Fourier
sin
( NII)
series
.
dn
trivial solution for the
P
pde
⑦
is
given
by
ucn.tl#Ane-kY-Itsin(nII)n-1
where
An
=
2-
[
flat
sin
( NII)
dn
@ Ramaiah University
of Applied Sciences
±___zs_fA--olT
--ÉÑ
Faculty of Mathematical and Physical Sciences
____--__f__p___fÉ-o
-
ff-o-sqknofw-aveegeeot-ionI.fi
c-
-
russet
.tn/--04uCL.t.--oUCn.01--ftn1&Y-t/t=j-gcN
Ufo
)
www.t/--AnCosCen-It)tBnSinCcnL-Tt)sinfnIY)n-1
where
An
I
-
!¥nsin( NII
[
÷
Bn=
/ die
gcns inlmt-fjdn.sc#nf-orh-eategeuelionkY-w,
¥
Ul it /
ul
n
,
01
=
o<
,
UCL 1-1=0
0
,
,
=
f- In )
nah
,
t >o
,
t
> o
022cL
.
P
ucn.tl#Ane-kY-Itsin(nII)n-1
where
An
:[
[
flu )sin(
NIL)dn
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__Lg-_↳÷A←
Faculty of Mathematical and Physical Sciences
-___-faf-_p___fÉ--
-
ff-o-F-xamphs-i.it
obtain the
kite
Jun
IN
U
soI
:
=
for the
A
:[
ucn.tl
-
Ane
N
where
UCI.t.to
①
U( 1. 1-1=0
,
NT
that foehn
solution
The
UCO.tk °
Jt
)
equation
0
I
UCM
Given
-
0
,
Ju
=
U( 0,1-1=0
of the heat
solution
'
equation
kl
0
)
=
x
-
t=i
&
heat
.
is
t.in/nIn )
①
l
-
L
An
.
2- /
flu )
( NII ) die
sin
0
flu )=Ñ
put
4 2=1
polynomial
An
in
lieu
,
"
An
.
at sinlnitn ) du
2)
Te
o
=
Tu
=
-
An
-
4-
substitute
-
+
cant
An
+
=
[ 21
"
-
u
/ f) v
'Hv
-
.
"¥¥" 1+4%1%-17'D
a
+
'
+
0
•%;÷§
:÷÷÷→]
%÷? ÷ ]
-
.
42=1
in
.
①
,
A
uol.tl
ulv
'
•"÷:
w¥É•
21 -1¥
-
tu
Guy ws÷¥4
=2fñ•s;;
-
+21¥? Jyp )
-
e-
" ""
tsincnñn )
n
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
-o---f-Ih-fTfI--ff--___
-
the solution
obtain
②
the heat
of
equation
1<8%5%7
U(
Ul 0,1-1=0
UCN
SOI
Given
:
)
0
.
L= IT
{
=
&
it
:{
ucn.tl
where
k
-
,
,={
.
☒
!
in
fcn )
-
-
0 < "
,
0
Ty
<
Me
CKC IT
,
,
is
sin
sin
/ NII )
①
( NII ) du
An
g
o
+¥inlnn1dn
Tin Cnn ) die
Tle
0
An
1
equation
/ ¥11T
2-
-
fall
F- /
-
IT/ 2
<
1172<22 IT
0
fin
< x
L
9
-
0
.
l
-
An
2=17
An
-
Ane
n
put
1
solution of the hurt
II
IT ,t 7--0
f- wsf.net/MIC--
¥
0
-
=
÷(
wscn
)
,
An
put
2=19
=
-
-
wscol
)
÷*(wsln÷) 1)
-
4 An
in
①
.
A
uol.tl
:[ :-, ( ws(n¥ I
-
I
/ e-
" "
tsinlnn )
nil
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__hI_f÷i&
Faculty of Mathematical and Physical Sciences
-___--__f__p_-→fÉ-o
ff--___
-
the solution of
obtain
③
i
the
equation
wave
:÷=i÷
Ulu -11=0
.
Ulm 07
,
UCH , t 7--0
,
It -0=0
:-X
.
sin
:
Given
-
the
that
f- cnn.se
solution for the
girl
,
wave
__
42
0
equation
-
given by
is
P
uol.tl
An
=
n
where
cost
-
I f. 4-cnisincn
-
÷
fcnt.se
2g
-
Bn since
) sinfn )
①
An
-
1-
-
,
An
)
l
-
Bn=
put
t
'
Idn
-20
fotgcnsinlnitejdn
4 tilt
②
in
.
sesinlnnldn
N
=
¥1
-
uwsin-ml-sin.sn# ]
ñws÷ñÉisinc;;
¥ I
=
put
-
1-
=
An
③
-
E- lutwsen-my-i.l-sin.in#j]
=
IT
-
-
÷
"
girl
:O
/
:# )
IT
÷
=
'T
0
'T
2=0
o
1
]
:
coscm-t.tn
sin
In
7
=
"
0
+1
till iii. 1- ii.
-
"
Bn
in
Bn
=
we
,
get
0
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
____-faf-_p_-→fÉ-o
ff--___
-
An 4 Bn
put
uln.tl
{
=
-
l
U
The
:{Ane
where
2x~
N,
01
kl
-
-
KOI
flat
A
-
equation
U
(
t.tl
=
O
Cos ✗
=
and
cosh
2=1
of the heat equation
solution
N
=
,
that
Given
ucn.tl
of the heat
10 -11=0
UL
-
Sin Inn )
)
co scent
¥
:
get
we
,
"
2
the solution
obtain
SOI
①
A
n
41
equation
in
t
sin
.
given by
is
/ NII )
①
l
L
An
substitute
An
2-
.
flat
.
=
2
!
fcn )
sin
( NII ) da
2=1
cosas
An
in
get
we
,
"
/
cosh Sin
200s A sin D= sin /A + B)
( NHK ) du
-
sin / A-
B)
0
/
=
incr + nm )
since
-
-
min
/
dn
)
)
sin
to ) :
-
since
0
/ ( sin ( linin )
=
+ sin
O
=[÷÷÷
-
( MIT
-
1)
N)
)
du
•"÷j
N
-
=
wscn
-
NIT -11
a.
"
=
-
•":
Faculty of Mathematical and Physical Sciences
-
ws!
•s1:
-0
'
l n÷+in÷ )
-
-
.
+
n÷+in÷
,
@ Ramaiah University of Applied Sciences
-___-faf-_p___fÉ-o
ff--___
-
An 4
put
L
-
①
in
l
-
,
it
aint ,
-
[ -001,7¥
N
⑤
!;+n÷+n÷
'
as
-
&¥=
tsinin.int
10 .tl
c2z2u
UCH -11=0
0
=
2¥ /
Ulna 7=0
lpiven
That
The solution of
equation
wave
Full
-
""
"
I
?
obtain solution of the
sod !
e-
,
"
f- into
the
,
gin
,
wave
Sinn
E-
+
,
=
L
4
Sinn
equation
is
-
-
IT
given
by
P
uol.tl
cost
An
=
n
where
-
An
-
1=0
&
=
put gents incnl
-
Bnsinccntt )
÷
Idn
sin
L= IT
③
-
egI②
in
,
0
t= -11
4
③
egf
in
sinoelsinl
,
Cn)-①
-20
fotgcnsinlnitfjdn
÷
An
Bn
It
Z f. 4-cnisincn
-
,
ten
n
l
Bn=
put
'
,
Idn
IT
Bn
=
÷
,
=
÷
!
Sinn Sin
( nm )
-
Hosen
-
nn
)
-
: 25in Asin B=os( A- D)
da
-
costa -1131
④
costa
inn
-
1) da
IT
=
÷ I @ ski
-
nm )
-
costa -1mn ) / du
i-g-og-zr-r-olg-FF.gs
Faculty of Mathematical and Physical Sciences
@ Ramaiah University of Applied Sciences
-o---faf-_p___fÉ-o
ff-o-tsn-en-rlo.in#-nn1-sinY::I
-
]
"
,
n
,
NIO
sinf.it#-sinY-+:I-o)Bn--0.n--1pntn->
☐
=
1
B,
¥ /
=
④
egf
in
,
"
Sinn Sinn
,
du
0
¥
=
¥2
÷
=
,
"
!
[
=
/
Sirin du
(
1-
u
-
2 sink
=
1-
wszu
wsanldn
sing]
'T
HI 0
[
=
B.
since
An
=
,
sin__
-
o
]
÷
=
0
IT
Bn
uol.tl
= 0
÷
uol.tl
=
,
nai
d-
4
=
since"¥ )
1- since
Faculty of Mathematical and Physical Sciences
⇐÷_-gf@_grdrggg_f_
B.
t)
t.lk equation
sin
(
Yf
①
implies
that
)
sincnl
@ Ramaiah University of Applied Sciences
-___-faf-_p___fI-ff--___
-
obtain Itu solution of the
⑥
ice
UCO.tl
=
2¥
c-
=
¥
,
UC
0
IT
,
f-
±
"
:
'
The
Given
(n
)
:{
that
sink
t)
Mz
{
feat
Sinn
←
CN
<
An
=
n
where
-
cost
is
An
+
given
Bn since
I f. 4-cnisincn
-
/
Idn
=
f- it
in
I
=
③
-
②
,ydn+ /
70
1T
sincnlsinln
du
0
Me
"
F÷
1-
sin
-20
gcn )sin(m¥)dn
°
=
)
i
÷ /
fins
G-
=
)
y
①
,
An
8th 1=0,42--7
'
<
l
Bn=
substitute
t
'
IT
11-12 < u
0
A
uol.tl
%
a- 172
• <
equation
wave
0
=
o< u
0
.
solution of the
,
¥+1T -5-0
Ucu 01=1-1^1
where
equation
wave
Sinn
sinlnn )
du
④
/
I
25in A
sin B-
WHA
-
B)
-
Costa + B)
-
"Fws(
!
!
n
-
na
)
-
coslntnnlldu
The
÷ / costa
=¥[
-
nin
)
-
cos
/ lunk ) /
dn
sina.IN#-sinH:;;-nJ*.n-ti
✗ =D
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__hE-_↳÷Ñ
Faculty of Mathematical and Physical Sciences
-o-_-faf-_p-TfI---
ff-o-n.in
[
-
sina.÷[IMsina.IT#-sinlY;:--H).n-+i
#-sinlYIn---l-o),n-tiAn-:#
-
put
in
n=1
A.
④,
Tina
It !
=
Sinn
÷ ! Tinta
=
!
÷
-
f- [
=
a
du
/
du
,
=
⑤
2
Sirin
.
I
-
cos an
wsanldn
Sinan]
-
Mr
N =D
A,
put
glue
:[ E- ¥
I
=
o
in
③
-
⑤
,
⑥ 4 ⑦
in
-
o
]
⑥
get
we
,
Bn=0
put
"
"
=
-
⑦
①
A
uol.tl
-
A ios (
IE) sin / II /
An
+
n
-
cost
t
'
)
2
-113m¥?
) sinfn )
b
Wnt I
=L coset ) Sinn
where
+
[ An
wsccnt )
sin Cnn
)
An=÷[sinT&mn" sin(¥;]
-
Term
Test
-
l
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-gho-fi.A
T
____-foIh-p___fÉ-o
ff--___
-
cha's method
① obtain
complete
the
yzp
5¥ "
of the pole
integral
?
q=o
-
①
www.derflqy.z.P.E/--Yzp2-z-1ODifftegf
partially
Let
us
w.at
①
foe
=
0
char pit
fy=
,
's
dp
auxiliary
d¥y
%÷
fq
given by
is
=
dz
=
=
,
-
zdpI-gg.pe
=
-1
pcgyzp, _gµ,
¥p
=
4-
.gg?F-+g---zdyn-zp--dY-
=
terms
& last
÷
¥
dy
-
,
-
Ipr
pt
-
>
Jj dr.
ydy
-
,
.
YJ
=
y
.
-
-
p:
a
:-#
¥
Y
¥ a
p
=
-
RE
⑦
in
- •
q
YI
az
p
put
,
fp=2yzp
,
equation
z¥÷ypi
=
>
1st
'
Psg
z
=fyd÷fz=→¥÷q=?÷= ¥5
empt
Take
fz=Yp
ZP ?
a. Y,
.
:
-
-
o
Ya÷
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
____--__f__p___fÉ-o
ff--___
-
dz=
WKT
da
✗
an
pdn
qdy
+
taFdn-IT@7f.yrra.y
-
Fda
É
dz
die
=
+
Ya¥u
Ya?Fdy=
-
-
dy
du
dlz.ra-yydfz.at/--dn--Ta./ y?dz+zk--ydy
integrate
2-
z
both sides
on
iay
Ta
-
-
y
adf.ir
,
b
N +
=
)
4=4+312
tankage's method
⑤TY*÷,+④%=⑨R (
say
:
-
The
auniliary
of
dg-z.dz
choose
% 2nd
is
pp
+
aq=R )
given by
¥
=
terms
,
d¥= dfa
fadn =/ydy
É=
au
=
!
y
y
II
+
÷
4- 4
'
=
C
,
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
____--__f__p___fÉ-o
ff--___
-
2nd
choose
ditzy
3rd
and
linn
d¥ey
.
)ydy=§dz
E- E -1¥
-
2=22+12
y
y
the
G.
s
.
she
'
-
-
for the given pole
is
.cz/=o@9--&ce-)&Cxt-y3y'-z7-#Type-2ofcc
④
se
,
(u)p+①yg=2④R
P
A. Ei
,
daily
choose
Q
Fay
=
2- z
2nd
-
4
srd
gdyzy
d÷z
=
terms
d¥z
=
/ %- =/ dz
buy
lnz
=
my
Choon
luce )
,
tnlzc )
=
y
+
,
=
1st
2C
&
,
2nd
c.
=
terms
H2
2-
-
¥
,
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__LI-_↳÷A←
Faculty of Mathematical and Physical Sciences
-___--__f__p_-→fÉ-o
ff--___
-
!÷⑤= Fay
!÷÷i
2 my dn
Fay
-
-
-
"
✗"
tyg
2%
d
(
a
-
dn
dy
a
=
guy du
dy
¥)
y
-
-
dy
-
=
If dy
-
dy
Inui ÷:)
-
-
o
-
dy
-
=
-
dy
y
dy
-
-
4¥ dy
Edy
-
tidy
tidy
integrate
Ey
÷
÷
-
=
:
.
G. s
.
&
-
=
+
y
y
y
+
-
Egg
-
tis + er
35.*
-
→
c-
Cz
is
(C
,
,
Cz 1
=
0
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__hE-_↳÷A
Faculty of Mathematical and Physical Sciences
____-foIh_p_-→fÉ-o
-
f -o-types-404-ypp-ln-iylq-2NZA.edu?y- di÷y
choose
multipliers
the
d÷z
-
.
-
z
,
-2
_zdn-zdy+d2_
-
2
(
n
y
-
)
z
-
i=
,
-
tada
-
tydy
( nty ) + 2h2
+
I4÷←dz=o
z-Zdu-zdytdz-Yn-zy-zu-y-2nk-zdn-zdy-dz-n.co
zdntzdy
dz=
tzds
bn
(
-
-
z
)
-1dg
Nt
=
=
ee
y
+ c
,
-14-19
en
2- =
2-
du
'
.
en
-14
"*
choose
Idn
multipliers
+ i.
dy
-
as
Ida
dy iz
-
"
z%
.
Fingaz;
dnt
/
ene
d"%÷;÷
dz=o
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__Lg-÷É&
Faculty of Mathematical and Physical Sciences
-oG--faf--@_--→I-ff--___
-
Y
n +
-
bul 2)
C
=
>
2=eN+?
d÷y=dY
In
n-÷y
.
y
-
dye
v-iudgn-n-n.IM
v
+
n
a
%
dd÷
ru
v +
=
ridge
-
u
g
=
t✓
.
-
v
"
"
=
=
n
/
'
it
:-. "÷
,÷
dv
tu
-
,
/¥
dr
,
tain 't v1
du
-1£ ;÷udr
-
tell
+
it
-
)
toed
=
n
tu
t
b
taricu-nl-tzmci-izl-t.se#-&
@ Ramaiah University of Applied Sciences
÷-g_Es_gEz-oh
T_f÷Ép•
?⃝
?⃝
Faculty of Mathematical and Physical Sciences
____--E-f__p___fI-o
-
og-o-YY-iayyp-ayq-xzsodn.cl?-yn-- dY-m
:
d÷
?÷yy=d÷
-
buy
tnz
=
bull , )
e
d÷m
-
.
:
-
,
-
-
luce )
,
loathing
Zy
Em
defer →t÷yT
.
put
&
your
love
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-ghog-IF.AE
-o---faf-_p_-→fÉ-o
-
ff-o-DIPY-g.gg
qz
-
-
ply
f-
.
fu
-
o
fy
+
g
-
=
-
"
@
g
+
c
g-
Pbr
z
-
-2¥ zig
-
=
2
=
+
qz
_d%y+z
qdg
,
? G
y
=
g-
-
:p
fp
-9ft §
=
p
[
dy→
=
II
-
z
orgy
d÷qy
=
=
@
-
=
,
g
-
+
-9¥ ¥É
pdp
y
p' g- fz
=
Fg
9¥
'
-
:c
,
-
gz
44-2
9÷E
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__hI-_↳÷Ñ
Faculty of Mathematical and Physical Sciences
____--__f__p___fÉ-o
ff--___
-
p
dz
pdn
=
da
da
FÉ
=
I
-1
dy
g
fE÷I
-
dnt
Ñ
=
.dz
¥÷
=
du
dnt
i÷ I
,
d
+
Hz dy
ily
a÷⇐
,
,
-
Iz dy
-
;÷gydy=
(Hiryu )
=
Fig
-
eidn
cidn
-
c
in
+
c-
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__hI-_÷÷A←
Faculty of Mathematical and Physical Sciences
-o----E-f__p___fI-o
og-o-F-n-gkingmadkmk.es
-
-
function £
①
complex
valued
②
Limits
continuity
③
Analytic function
④
,
complex
line
-
and
⇐
Exponential trigonometric polynomial
4
logarithmic functions )
Cauchy
,
,
-
Riemann
equation
integral
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__l0-_↳÷A←
Faculty of Mathematical and Physical Sciences
____-faf-_p___fÉ-o
ff--___
-
Analysis
Empty
complex b
A
where
and
a
number
the real
The
*
The (a)
iff
Conjugate
conjugate
re
=
,
→
in
3
=
number
denoted
by
conjugate
of
z
,
-12
-
m+iy
=
then
I
then
i
In
=
2i+zT
,
→
the
=
=
21-+2
,
=
+
unit
imaginary
2
,
-_
.
( Rech at
-_
( Insta) b)
-
nntiyr
=
Z
2
are
equal
.
the number
,
6
=
I
then
-
2-
-
=
-
is
obtained
called
complex
.
is
a
-
iy
si
-3
-
si
Z
,
2
z
Imlza )
=
22--42 + IN
4
s
its
i=F
i.
-
-
iy
3
-
=
2
,
-
?
2- a +
part of
imaginary past
of
2=6+3 i
z
2
n
(2)
complex
a
z=a+iy
2- =
If
Im
i
outing
=
,
conjugate
e.
If
z
sign of
Itu
or
i.
-
satisfies
i
&
is
z
the
i is
imaginary
is
Re ( Zz)
and
is
unit
If
:
by changing
F- ✗ :
numbers
complex numbers
=
number of the foam
a
called real part of
called
a
b
imaginary
The Two
*
is
real
are
The real number
*
*
number
=
q-iyyzi-nz-IY.rs
native
icy -1W)
uz ) +
+ Nz
)
-
ich -1421
m-iy-m-iy.ie
I
+
Is
yr
→
=
✗+
iy
✓
y
:
-
2
"
¥
,
Faculty of Mathematical and Physical Sciences
-
-
as
-
ga
-
-
-
:
position
>
a
vector
@ Ramaiah University of Applied Sciences
-___-faf-_p___fÉ-o
ff--___
-
Moduhy_
modulus
The
is
absolute value of
or
given by
121 =
n2+y
denoted
z=n+iy
121
Ey
Ir
=
n'+ y
121
i.
121
*
=
it
uh-17218
=
*
'
=r
121
-
-
circle with radius
radius
circle
r
121 >
'
-
*
.
.
j
-
Er
-
.
Disc
r
.
121
→
r
↳
*
r
-
→
disc with
a
¥
=
'
represents
represents
Er
r
"
121
-
→
☒
by
,
-
-
Petar forms
coordinate In
Rectangular
y ,
,
and
Y
polar
^
•
coordinate ( r,o )
related
by
the
r
equations
2=n+iy
:
Itu complex
iy-rsinon-ircoeogy-a.se
are
Sino
member
z
z
the
raw +
=
z
angle
iy
✗+
=
.
Thes fore
r
Sind
rccoiotisino )
=
0
is
D=
called
avg.cz )
arglzl should
This
The
-
it
be written
can
i
argument
<
o
c- IT
.
of
is
an
→
The
I
.
.
n
as
22+44
complex number
argument
of
z
and
is
.
-
i.
>
rcoso
-
r
a
-
in
polar form
.
denoted
by
.
satisfy
tano
-
In
.
complex number in the
called principal
argument of
Itu
interval
z
.
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__h
T-f÷ÉÑ
Faculty of Mathematical and Physical Sciences
-o---faf-_p_-→fI-ff--___
-
Ex :
1)
.
Obtain
sold
2
argli )
i
=
U
tano
argci )
2)
obtain
sod
Y
4
0
=
iy
✗ +
=
1-
-
→
,
i
+
n
-
tune
n+
=
4 y
l
-
o
I
=
asgci
2--1
1
=
+
-
I
=
tano
③
empress
so
I:
I
n
-
r
g
-
-
Fsi
l
-121
-
tano
y
-
-
In
1- and
o
-
.
-
-
-
Cosa
principal
polar form
=
=
-
Ia
→
B
¥2
=
-
Ig
=
in
4
-
I
-
Sino
O
ti )
principal argument
iy
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i
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sin
polar
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given by
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@ Ramaiah University of Applied Sciences
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FLEM
of comin variably
off
range
iy=g w=t%
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a-
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Domain
off
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valued
function f
function
auigns
the
B
is
ut
element
each
to
W
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flz
7
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are
F-
complex
An -2^-1 an
non
function
i.
,
function
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=
2
+
YmPhk
k
.
defined
flag
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,
2h
-
'
+
-
-
+
.
is
by
and
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enponential function
=
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e
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efficient
polynomial
a
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"
e
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an
an 2 f- ow
of
fcz ,
°
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Ai
degree
is
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i :O, i. 2.
n
defined
.
by
( cosy + isiny )
t-garithm.ie
function
is
inverse
enponential
of
.
e
in
functions
:
w
=
lnz
-
②
only
'
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f- (2) = e-
-
complex
a
that
correspondence
and
element
,
negative called
integer
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a
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Logarithmic
one
z=n+iy
function
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:
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complex number
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f
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w
from
vt-vca.nl
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plane
uol.nl
2
=
①
if
=
-
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ew
-
ew
Ntiy
z
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ell eiv
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|
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✗ +
iy
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n
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zetiy
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me
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③
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eu
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y
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of
to
is
z
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unique
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,
i.
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In
=
n=
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that
.
0
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the
since
,
if
is
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,
is the
o
argument
argument
of
2
lnz
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=
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I
(0+2^7)
n=0 ,
,
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,
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e
the
For
Logarithmic of
a
-1-0
and
,
bnz
=
then
that
w
=
lnz
i.
,
0 -12nF
implies
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argcz )
a-
a
complex
number
is
given
arglz )
logelzlti ( 0+2
NIT
/
n
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-0 -1-1
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,
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.
-
by
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Limit
of
function
a
Vr
D
Su
Sa
plane
2-
w
-
plane
suppose the function flat is defined in some neighborhood of
possibly at so then f is said to be possess a limit at a
,
him fed
so
,
except
written
as
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z -520
for each
if
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then
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=L , L2
glz
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=
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function
La -1-0
,
.
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conli-m.ly
A
L,
=
glz )
-520
him
3)
.
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him
2)
him
and
=L ,
z -7 20
1)
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whenever
c-
<
nee that
or > o
limits
of
limflzl
suppose
L
enist
there
,
aka point
fcz )
limfczl
z→
said to be
is
=
continuous
at
f- too )
D
zo
Derivative : suppose the complex function f- is defined
The derivative
of f at zo is
of
point
a
20
.
'
9- to )=
him ffzo
oz
provided
this
so
-102
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oz
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limit exist
if
2--2
.
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flzo )
in
a
neighborhood
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the limit
If
in
the above
function fcs )
the
that
EI:-p show
sold
wz
f- (
=
-1021
z
-
+
a -14
=
flat
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so
along
Oz → o
fk+°Yz-f
him
i.
let
we
along
fear
=
not
is
n+4iy
Dz
r
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-0
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÷
ioy
to x-axis
then
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oy
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s
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say
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at
nowhere
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iy
then
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to
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is
flat
on
equation
So
differentiable
y
-
anis
then
on -0
-
4
=
at
.
any point
z
.
A¥tic function
complex function
A
at
point
a
point
in
A
at
-
Itu
if f-
is
in
-
-
flat
said
is
differentiable
neighborhood
function flat
every
The
zo
w
of
is
zo
at zoos at
analytic
condition for
-
-
analytic
every
•
a
.
in
a
domain
D
↳
D.
necessary
be
to
if
flat
is
analytic
analytiiihi
-
cautery Rien equations
-
differentiable at z=n+iy Then
It first order
at
partial derivatives of u and verist and
satisfy the Cauchy Riemann equation
suppose fcz )
=
ceca
,y ,
+
ivcu y )
,
is
.
z
-
2÷= Ey
and
Fy
-
-
Fn
.
Exampli
:
the
show that
1)
analytic
at
point
any
that
Given
function ftd-12nh-yj-i.ly In )
is
not
.
,y1=2Ñ+y
vcn.is/-.yIuDiff-t
partially
SOI :
and
u
÷n
%
observe
ucn
In
an
.
the
Cauchy
function
Ey
+
flat
1- to
-
l
::•÷→y
24
=
Given
Difft
that
¥y
=
=
'
-
y
-
In
satisfies
does 't
analytic
.
therefore
.
Riemann conditions for the function
+
a)
ucn,y ,
=
partially
Uav
Ya
-
-
equations
not
is
In
=
Fy
←
Riemann
-
verify Cauchy
SIM :
-
-
y
that
In
②
-
Ey
1
=
and
w.r.tn
v
-
and
+
/ My + y )
i
n'
-
w.at
+ n
Fy
-27
vinyl
.
2mg
+
nay
.
In
2n +1
and
-
y
2g
-
=
2n -11
observe that
It
In
Cauchy
-
=
Ey
=
Riemann
2^+1
s
Koy
.
-
equations satisfies
In
.
=
-
sy
y
the
the
verify
③
Cauchy
f- (2)
SIN
!
Given
-
-
322+52-6
=
that
equations
Riemann
i
f. (2) =3 ( ntiy )
=3
+
( n' + i'y
Csu
=
"
'
-3g
2=2
,
7121=322+52
-
Gi
+
su
)
Ntiy
-
+
function
iy
+
-6 i
ginytesniisy
-
+
the
2-
,
slntiy )
-
for
i
( cony
-
+5g
-
Gi
it
,
-
6)
UCN.ly/--3N~-3yt-jpartially
u,VCn,Y1--6Hy-5y-GDiff-t
i.
and
u
¥n
In
the
He
6N -15
.
Fy
w.r.t.ua y
v
-
=
Ing
=
Cauchy
-
%y
by
and
Gn -15
=
Riemann
by
=
6k -15
=
tug
equations
In
-
=
=
-
verified
are
by
.
czi-kri-onf-oranalyti-u.ly
suppose the
real
valued
uol.gl and
vcn.gg
first order partial derivatives in
uol.gl and vinyl satisfies the Cauchy Riemann
have continuous
and
If
-
point
of
D
analytic
D
E
in
complex function
the
then
,
f- (2)
is
functions
uln.ly/+iVln y )
=
,
D.
analytiu.li
It
check
fcz1=u÷F
sots
:
-
Here
ucn.gl
.
÷y
-
of Itn
i
function
a÷y~
and
vlMY1=
,
Difft
u
and
v
partially
w.r.tn
.
ay
_n¥-yu
are
a
continuous
domain
D
equations
.
at all
I
?⃝
UH y
÷=
-
µ :* ,
'
In
+
-
*Y
Y÷÷
-
,
nosy ) even
,
and
y,
equations
fcz )
2=0
,
4+441-11+24 !
=
civil
y
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@ Kye,
n'+5=0
satisfies
in
derivatives
partial
its
analytic
is
;÷÷ :-#
4-+442
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~
÷y=-:÷=→m_
( 24442
the point where
at
function
containing
•
.
+
-
¥
fu4yy~
Riemann
-
÷
-2mg
ye
erupt
Cauchy
The
y
ng
-
continuous
i.
-
=
functions
also
;÷÷
.
-
HIM "
=
F.
The
2kt
=
JU
guy
'
at
2=0
point except
all
domain
any
are
D
at
z=o
.
not
D
.
0,07
function
show that the
2)
SOI
!
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put z=n+iy
f- (2) =
in
since
Sinn
=
f (2)
Ula
Difft
2¥
¥
=
C
iy)
cosciy]
cushy
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+
flat
.
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icosnsinhy
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v
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coshn
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in
-1
-
isinhn
nay
.
sign
R
cos
cosnsinfiy ,
sinnsinhy
-
analytic
sin / At B)= sin AWSB
+
try
=
is
get
eosncoshy
25m
Therefore
Sinn
we
,
sinz
.ly/--sinncoshy4VCn.y1=cosnsinhycontinuous functions
-
and
u
=
ft)
ft ) :
=
Ing
=
oosaecoshy
equations
sins
is
=
are
=
9
-
sinus
¥ ¥n=
satisfied
analytic
.
;nhy}
cosncoshy
=
-
and
continents
given
in
domain
any
sinnsinhy
hence the function
D.
-o-_--__f__p_-→fÉ-o
ff-o-c-R-yinpo-ar.co
-
ordinator
¥ Go
Fo
r%
-
and
-
-
-
-
-
③
that
show
sold !
f- (2)
-
put
reio
z=
flat
logz
=
7121
in
log / reio )
-
Difft
I.
¥
r
-
logo
and
u
and
V=
÷
÷
=
0
w.r.t.ro
-1
-
.
%
-0
-
=
v.
f-
-
-
1=40
=
rYg=0= ngo
function
c-
is
Regeuetions
analytic
o
•
Fr
r%=
-
therefore
get
logrtio
partially
v
.
+
=
U
we
,
analytic
login logeio
bogrtiologe
=
=
=)
is
2-+0
,
are
satisfied
¥
-
-
Koo
and
hence
given
.
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-gho-fi.A
T
_o-_--E-f__p-Tf---→g→--___
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Hazmi functions
A real valued function okay ) is
if it's second order partial derivatives
satisfies the Laplace equation
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suppose f- 121
Then the functions
Hankie
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such
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sohf
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ivcn y )
,
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function
function
very ,
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domain
functions
☐
.
domain D then
that is harmonic in D
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a
analytic function
function of
conjugate harmonic
is
.
an
.
is
U
.
harmonic
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uln y )
,
partially
3ñ→y~
=
-
=
✓
II
Ky
&
Gre -6N
function
=
the entire
=
-
Ji
y
Guy -5-10
-
Gu
Continuous in the
entire complex plane
→
0
way ,
Therefore the function
,
w.r-t.ae
Fu
s
Gu
-
-1
in
in
function ucn.ly/-.a?3xyt-s-y
entire
complex plane
the
conjugate harmonic function of u
Difft
,
the
analytic
harmonic
.
%÷
"
+
is
that the
the
true
-
.
,
the harmonic
is
is
Find
a)
,
find
ucn, y ,
Verify
-
uca
,
always
b)
-
g) + ivcny )
y ) and vcn.gg
ceca
=
uln y )
in
continuous and
o
-
conjugate
vinyl
a)
are
harmonic function
function
Sifu of
suppose
said to be
satisfies Laplace
ucn y )
complex plane
,
is
equation
harmonic
.
function
.
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__h
T-_↳÷A
Faculty of Mathematical and Physical Sciences
-o-_--E-f__p___fI-gÉ-g _
-
b)
the harmonic
since
c-
equations
R
and
÷g=s÷
From
①
&
vinyl must
conjugate
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-
-
,
Fy
In
Integrate ¥y partially
3k13g
'
&
-
=
w.at
3dg
-
3y÷
-
-
V
Diff
3N
-
t
-
>
y
-
Guy
=
,
From
y
③ &⑨
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w
h Cn
integrate
)
The
,
y,
=
r
-
t
=
6mg + 5-
③
y
Integrating
constant
④
a
.
-
⑤
5
both sides
on
=
-
-
-
only -15
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1- Guy 5)
have
'
④
+
h1n1
+
we
,
hen ,
+
partially
v
¥
②
②
=
v
satisfy
n
3n2y
analytic function
+ c
-
y
>
+ su + e.
→
Required
conjugate
harmonic
of
U
.
is
fczt.se?3nyt-s-y+i(3riy-YJ-i5N- c)
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__lT-_↳÷A
Faculty of Mathematical and Physical Sciences
____--E-f__p___fI-o
gÉ--___
-
②
b) Find
Ula
,
ykécawsy
analytic function
:'
Fu
Difft
=
Mtn
-
ysiny )
e
( 2 wsy
"
,
partially
u
=
Mjg
Ing
e
"
writ
form
ey
ysiny )
-
nary
-
-
=
e
( any
"
+
away
-
ysiny )
ysiny )
ysiny )
y
.
-
l
-
e
-
"
,
=
e
-
-
"
0¥ -10¥
using
(
→
-
C-
y cosy
cosy cosy
-
C- awry
en
Ju
a
cosy
-
-
-
sing )
C- ysiny
-
way
+
awry
y sing
+
+
-
way )
)
cosy 1
y sing )
ey2wyyi-uwy-yiny-uwy-seoiy-ys.my)
Laplace equation
III. +0¥
,
=
,
=
therefore
+
cosy
+ a
and hence
C- using (young +sing 1)
en
=
gyu
en ca cosy
+
u
.
w.r.tn
emory + e4 any
=
Difft
Ky
en cosy
=
idea
partially
a
function of
harmonic
conjugate
the
the
:
function
harmonic
is
six
the
that
Verify
a)
.
The
o
function
→
ucn y ,
,
is
harmonic
function
.
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__l0-_↳÷A
Faculty of Mathematical and Physical Sciences
____-f-Ih_p_-→fI-fÉ-g _
-
the harmonic
since
conjugate
=e"( any
%= teal
%
c-
R
equations
%; Fy
-
many
using
-
-
+
satisfy
must
u
and
Eg !
Fy
of
-
-
ysiny )
ywsy
-
①
)
sing )
e4nsiny-ywsy-isinyj-s@Integrali Opartial yw.r.t
=
,
luv.ie/v-uYIvv--eNsiny-neksiny-en(yc-wsy?+sinyji-inln)vy
.
eysiny-uensiny-ehycoey-einy-hcnlv-xeksiny-e.my
Diff
partially
v
÷n=
From
②
w.at
.
u
sing ( neuter )
④
&
encasing
③
then )
cosy
+
éywsy
+
hYn )
-
④
have
we
,
+
ycosyi-sinyj-sinycaek-e.tl) ékywsy -1h 'm
+
)
seeming éywy eying .net#y+ensiy-e/ywy-inYn
+
+
-
)
n' into
integration
h1n1
Equation
The
flake
③
both
=
sides
,
C
vcqy ,
=
mensing ernywsy
+
corresponding analytic
"
Gewsy
-
ysiny )
+
ie
"
function
( using
+
-1C
is
goosy /
+ e
@ Ramaiah University of Applied Sciences
÷-g_Es__-ng__L
T-_↳÷A
Faculty of Mathematical and Physical Sciences
-o---f__f__p___fI-o
gÉ-g _
-
bogecrity
Klay / =
)
'
harmonic
is
b) Find
the
say
function
the
that
Verify
③ a)
:
the
analytic function
Dittt
-
harmonic
conjugate
u
partially
%n=
Y÷=
w.r-t.se
"
)
-
4444121
-
arm
city -5
÷=÷::÷
7:÷→
*
x4y
Igy
"÷÷÷
she
form
%
=
;
@ 2+55
=
and hence
& y
Ef
s
,
4:44 121 -242
u
.
Ju
nay
function of
÷i÷÷g :
Take
"÷i+¥=?¥:¥+;÷÷E=ax→K+m→x
@ 2+425
0¥ -10¥
therefore
b)
since
c-
R
the
=
→
o
Laplace equation
,
,
function
the harmonic
conjugate
equations
Ey=¥n
%;
&
5- :÷
r
=
-
Ucny )
①
-
is
harmonic
of
ufiev )
must
satisfy
Ey
:÷=→_
a'+ y
-
@
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__hT-_↳÷A←
Faculty of Mathematical and Physical Sciences
____--E-f__p-Tf---→og--___
-
Integrate
partially w.tt
tetanic In /
①
2h
✓=
+
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③
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tical
integrate
=
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is
the
=
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is
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both sides
=
Yu
,
,
C
atari ( Hn )
required
login !iy4
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harmonic
corresponding analytic
flat
h
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on
h (a)
have
We
.
+
function
i 2 tan
'
conjugate
⑨
of
u
.
is
( Mn
)
+ c
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__lT-_↳÷A
Faculty of Mathematical and Physical Sciences
-o-_--__f__p___fÉ-o
ff--___
-
Moti
:
.
always
can
Suppose
way ,
find
harmonic
its
analytic wing
is
complex
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or
is
c
curve
integral
§ f- (2)
integral
a- Nlt )
dz
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defined
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a
-
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path
or
eloÑ
integral
contour
point of
at
# c- b. The
flat dz
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11PM
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If f
ankur
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continuous
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a
integral
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smooth
contour
UP 11-50
2k
,
by / ftzldz
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n
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ucn
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smooth
off
integral
defined
along
c
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.
by
is
02k
1<=1
=
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Max
,
F-
-
,
§
where
curve
a
as
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denoted
is
if Itu
-
,
contour
c
on
Integral
be
f-
-
Fy
s
piecewise
a
,
ako called
of ftd
Cankar
Let
variables
referred
is
conjugate
integrations
coup
In
Ey
=
function
harmonic
given
equations
c- R
F.
is
integral
given by
zltt-ncti-iyltl.az
then
It
f flzldz / fczlt 1)
is
a
on
smooth
t c- b ,
b
=
z
C
1
sold
It c-
:
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4
't 1)
-
a
Evaluate
F- xam-pbi.tl
-
C
weave
/
where
I dz
c
is
given by
rest
,
g-
t
'
C
.
Given
n ist
4
-
4--1-2
ziti
'
z
f- 121--5
Therefore
=
,
3-1
+
it
~
=
It
-
It
it
{ fczldz =/ bfczct
-
ntiy
=3 t
+
it
-
1=3+1-2 t
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a
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Faculty of Mathematical and Physical Sciences
@ Ramaiah University of Applied Sciences
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4
/ fczldz
/
=
C
(3T
-
/ ( 3+2 it / dt
'
it
i
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YYqt-c.it
=
-
sit
-
+
2Ñ)dt
4
1,191-+21-3
=
+
=/ ÷ -14¥/
90¥
72
=
/
1-2.4
128
+
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-
5
195
I dz=
+
"
I
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4
dt
4
(
=
sit
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1¥ !
(%
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↳
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2)
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i
(43+1)
i 65
ist
-
c
§ tdz
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②
where
,
c
is
Itu circle
n=
cost
,
y
-
Sint
,
c
0kt
golf
:
.
EAT
Given
f- 121=1-2
Therefore
,
.
n
-
cost
e-
=
y
4
241
Sint
-
=
cost
+
is int
=
eit
zllti-ie.it
it
b
§f(
2)
dz
/ fczct
=
)
/ ziltldt
a
c
text
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=/
if
=
dt
0
=
i
(f)
21T
0
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2mi
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obtain
③
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-
z
where
dz
is
c
by
given
,
211-1=3++2 it , -2<-1-2-2
.
C
SOI
!
-
Given
f (2)
=
ziti
2-
-
,
✗ 1-
=
2^11-1=3+25
3t+i2t
+
i
,
21-5=91-2
at
-
-
+
i
12 t
-
b
therefore
/ fczldz =/
flat )
) a' ltidt
a
e
2
/
=
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19th
at
-
-
-
+
i
121-413+21-1
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i 36T
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formed
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be the
c
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analytic
cloud
fcz )
in
is
iimple
Z
-
connected domain ☐ 4 ht
within the domain D.
trying entirely
curve
§ flat
limply
a
dz
where
gmtiflzo )
=
Zo
within
C
Evaluate
1)
Ñ-4Z
§
dz
I
z +
where
,
¥
Let
us
and
take
Zo
f. (2)
-
-
-
Z +
C
22-42-14
10in
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21T
121=2
g.
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iflzo )
-
-
•
z
i
&"
/z
i
21T
=
dz
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)
i
z
( 1- if 4C it -14)
21T
if -1 ai 4)
=
§
if
2ñi
=
(
2mi
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3
+
,
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z
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/
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radius
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+
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zi
i
z-Edz=
§
=
.
the circle
is
c
point
a
C
e
•
is
20
ai )
/ z al
-
-
r
c
§÷+gd=2
Evaluate
21
,
where
c
is
the circle
12-21-1--4
.
si
C
•
±
"
"
'+g=(z+ , ;)
z
E-
point
but
f- (2)
lies
si
-
,,
imide
.
,
Itu
circle
bins outside
si
42-20
z+zi
=
gi
the circle
2
=
3T
•
2-3
i
•
¢21
•
go ,
•
-
§
c
2-
da
:
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§ ÷gd2 /
•
-
gi
-3 i
20--31
c
=
=
§¥g
21T
if (a)
anti
da
=
(1+7)=2 if }÷i)=-•÷=-÷÷
-
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in
C
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-E-fi.A
T
____-faf-_pTfI--ff--___
-
Evaluate
3)
the
shy !
-
Circle
121--3
-2
=
2+21
ai
+
%
i
•
theorem
By Cauchy integral
2
zo
-
-
f- Go )
21T i
=
is
e
.si
-
c
z
=
i
2-2-34+1.dz
where
.
.
7121=2^-32
Let
dz
'
c
zu
§
£→Z+
§
integral
An
-
=
-12 i
=
=
=
•
,
.
zi
2ñi( 202-320+4 i
2mi
21T
((
→i
if -4
Ziti
5- 31
Gi
+
)
) -14 if
ai
-
+
ai )
f- a -110 i /
Guarded candy intyI theorem
fcz) be the analytic function
he e be the cloned curve lies
Let
and
§
c
1)
Evaluate
HI
(z zoynti
§
e
:
-
n
2
d2
24+423
entirely
is
c
within
•
;
!
where
the
2-11
=
/
20
=
0
n
,
point
C.
121=1
.
-.÷
::÷=;-
f. (2)
☐
then
D.
circle
.
domain
the interior
is
of
,
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connected
limply
a
&" 7%20 )
-
-
iw.in
-
dz
in
-
-1-1=3
i
2
n=
2+4
§
ZI
24-1423
dz
=
§(Z+zY}-
for
%÷
=
=
=
Hi
-
3¥
,
dz
ti
e
c
-
'
-
=
"
f to )
=
_µ÷p
-61¥
g g- F' (2)
"
"
"
+ to
.
-
-
F'
=
"
312+452
¥4,5
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-←÷p
-
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c- ' =
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__hE-f÷i&
Faculty of Mathematical and Physical Sciences
-___-faf-_p-TfI---
-
ff-o-H.in
following integrals
Evaluate the
.
§ z÷i
2)
§
3)
§
5)
%%-
12
:
dz
:
/
i
-
2-
1--2
2) =p
121--1
;
f. ¥ ,zdz
41
-
£÷dz
121--4
:
§ z¥2d2
4)
sold :
dz
;
c->
12-21=1
,
SOI
:
-
¥ ,i"¥=¥¥n+
fczl
=
É
20--1
7^121=-32-4
§
C
g
,
F' (2)
=
n
:
,
-11=2--7
n
:
I:o)
s
I
-32
,
zt-p.dz:2?,i-f'Go)--2Hi-g?-=2itiC-
§¥pdz=
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-
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@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-gho-fi.A
T
____--E-f__p___fI-o
og-o-pzebmabik.ly
-
and statistics
operations
sent
ht
A
AUB
An B
B
{
re
=
/
&
then
AA
☒
KEA
NEA
a
=
Complement
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two
are
:{ /
AND
If
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and
U
A-
{
=
of
a
U
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/
a c-
An Ñ
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ult
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a
or
and
B)
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a
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devoted
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A-
u
°
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A- n B-
B-
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of
A
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☒
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ids
given by
is
sect A }
its
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disjoint
are
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Avis
De
Morgan
Laws
's
probability
deterministic enpesiment
whole outcomes
result is known
☒
A
or
*
A
is
with
certainly
.
random
probabilistic
experiment
whom outcomes
result
or
is
or
can't
*
be
predicted
Sample space
out
of
all
with
s
pouible
of
not
certainly
a
random
outcomes
experiment
an
of
is
an
enperiment
unique
s
ther fore
.
enperiment is
the
experiment
the
.
@ Ramaiah University of Applied Sciences
÷-_-Es_EEz-oL
T-_↳÷A
Faculty of Mathematical and Physical Sciences
____--E-f__p-Tf---→fÉ-g _
-
Event is
*
events
Two
*
liebert of
a
and
A
An
i. e
*
B
list of events A
,
exhaustive if
Ñ Ai
=
A
space
.
exclusive if
mutually
are
happen simultaneously
B
can't
sample
a
A
and B
of
A-
.
-
-
,
An
are
said
to be
collectively
s
=
i :|
The
*
If
☒
entire sample
space
event F-
an
ways
PCE
defined
by
and
B
In
Ai Ai
.
-
-
,
PCA
,
or
A.
or
p¥t theorem
If
-
-
on
is
defined
likely
by
calls
=
of
I
-
F-
In
denoted
=
1-
by
PCÉ )
PCE )
1
=
mutually
)
pl A)
=
An
An
PIA
then
exclusive events
then
PCB)
mutually
are
=
+
exclusive events
)
,
+
play
+
-
-
-
+
plan )
probability
of
A and
PCA
-
PLÉ )
two
B)
or
n÷
=
t
are
generally
If
pouible
and
equally
of
probability
of
PCA
event
for E
cans
occurrence
non
p (E)
A
out
by PCE))
E
of
Total
p( E)
If
universal
=
of
Addition theorem
a
ways
on
Favourable
my
=
probability
The
is
)
called
exhaustive
collectively
( denoted
,
probability
then
happen
can
exclusive
mutually
is
s
B
and B)
are
=
independent
events
then
PCA ) PCB )
-
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__lT-_↳÷A
Faculty of Mathematical and Physical Sciences
____-f__f__p___fI-o
gf-o-n-d-d.tw
-
route
n
PCAU B) =p(A)
☒
+
p( B)
-
a
PLAN B)
S
B
Ants
a-NB
§
8
A and
It
*
then
B
plan B)
mutually
are
exclusive
0
=
pl AUB ) =p /A) + pl B)
conned probability
Let
of event B
conditional
A and
when
the event A is
p( B) a)
"
pl
=
already
probability
P¥?
=
-
•
If
-
A
①
and
B
xam#
plan B)
1)
PIAU B)
say
:
If
=
WKT
WKT
A
314
of both
occurrence
of Itn
pl
=
and
B
B)
-
A and D
given
event A
•
PIAIB )
①
events then
PCB/ A) =p /B)
PIA )
-
are
events
B)
=
with
PIAF 318 pl B)
PCB/A)
,
p( A / B)
PlAp%÷
PIAU B) =p / A)
plan
=p (B)
=p (A) PCB)
then find
PLAID )=
B)
independent
two
are
plan B)
F.
.
occurrence
plan
plan B) =p / B) PLA / B)
NIK :
of
is called
happened
An
of
happening
the
probability
Plp÷¥
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the
.
by PCB/ A)
probability of the
,
PCB/ A)
111
denoted
probability
•
Itu two event
be
B
+
p / B)
4
-
-
=
518
&
.
①
plan B)
plait PCB )
-
pl AUB )
-
②
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__lT-_↳÷A←
Faculty of Mathematical and Physical Sciences
-___-f__f__p_-→fI-gÉ-g _
-
From
②
① ↳
,
p(
/ B)
A
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plans ]
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+
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-
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pl B) A)
2)
Givin
p( AUB )
sold !
,
%÷=
Plan B)
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play
Is
=
F- 1✗
=
g-
Is
=
p (A)
#
¥0
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¥
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÷
=
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=
314 PIB /
=
,
plan 5)
PLÑAB )
,
1154 Plan B)
PLA / B)
,
,
=
PCBIAI
PIÑIB)
,
find
then
1120
,
PIA / B- )
.
-
i)
p( AUB ) :p (A) + PIB )
3-4+1-5
=
PCAUB
ii )
plants )
=
=
)=
-
plan B)
-
I.
'
5+4-1
=
=
20
18-20--9-0
%
PIA )
3-4
-
-
plan B)
AANI
s
☐
§
a-NB
8
Io
sj-r.E ?jplAnI)- ? oi 1p(A-nB1-.plB-plAnB)- ' s- Lo'
=
%j-
p( An B)
=
Faculty of Mathematical and Physical Sciences
⇐÷_-gff_grdrggfggf
=
So
Igo
@ Ramaiah University of Applied Sciences
____-faf-_p___fI-ff--___
-
in
plat B)
P¥?¥
=
PIA / B)
v1
"
l
=
=
Igo
=
÷
¥
t.PH#fY- -Y;Ya- -Ey&- !-plBlA)- 'TsplAn-D)- plAuB)
plñ1I)=Ppñ;
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Plñ / is
plans
)
-
De
-
Morgan
=
'
I
f-
)=P';f☐
PLA / I )=
¥
's tar
=
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l
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PINA
=
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=
111h
=
111h
Find
7¥
PLI / A)
=
?
-
-
-
-
,, ,,
Find
PCB / A- I
÷
=
¥:#
=
¥
?
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-ghf-fi.A
T
-o---foII-p___fI-ff--___
-
Baynes theorem
congeal
on
probability
of exhaustive and mutually
exclusive events of the sample
each
space s with PCA ;) -1-0 for
If A is any other event www.ated with Ai (Ac Gail with
Let
Ai Az
.
.
,
be the
An
.
ut
i.
.
then
PIA 7=10
p(
Ai
in
/ A)
PYHPLAIAI E.pl
=
)p CAI
Ai
Ai )
i:|
1)
Three
machines
A
and
B
,
produce
of
factory
60%
C
of the total number of items
defective
output of there
item ulected at random and
that the item was
An
probability
only :
an
item
i.
suppose
PCD/ A)
3%
,
,
A
B. and
,
from machines
PCA
1=
60-1
the
D
is
=
2%
event
÷
probability
=
,
find 1£
from machine
By Baye 's
p ( CID )
c
we
,
o
=
theorem
PCB)
6
PCD / B)
that
of
zupectively
.
Find the
.
e.
of selection of
3,00-0--0.3
=
=
%
,
defective
a
O
-
plc )
-
03
to
item
PCD/ c)
ulected item
find plc / 1
a
is
:
=
=
o
-
I
them
¥
0.04
:
produced
have
we
,
Plc )
=
PC A) PCD/ A)
10%
se
.
need
a %
machine
events
the
of selection of
0.02
-
.
To
As B.
6¥
=
.
for
stand
c
g
and
found detective
is
produced by
Let
-
2%
are
.
,
percentage
The
a
machines
30%
,
P( B) pl D)B)
1-
0
.
PCD/ c)
.
I
✗
+
pcc ) p (D) c)
-
0.04
=
0
6
.
O
'
✗
Or 02
0.3
=
0
=
0.03 -1 -0
✗
,
I
0
Or 04
00 4
=
PCC / D)
1-
.
025
0.16
0
.
I 6
probability of the elected
( is produced from machine cis
The
(16%1)
item
016
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__l0-_↳÷A←
Faculty of Mathematical and Physical Sciences
-o---foII-p___fI-fÉ-g _
-
four machines
factory there
production
percentage
rupee lively of Itu total production
2) In a
machines
is
20%
,
there
Their
& D.
25-1,440-1
was
are
,
a
.
,
item
-
4% , 3%
5%
defective If
item is drawn at random
found defective then what is the probability that
D ?
was manufactured
by A
2%
Solin :
15%
,
out
.
and
A. B. c.
are
the
or
that
Given
-
PIA 1--2%0
Let
p(
✗
be
✗
/ A)
probability
find
To
by
Aor
D
and
A
Since
PLAUD / X )
By Baye
Pl
A
/✗)
find
to
PLAIN
=
0.15
20%
,
Pcc )
,
15%-25-1.4401
:
,
's theorem
-1
item
PCD) :
,
÷
"
.
-0.4
-
then
eucluiive
PLD/ ✗ I
manufactured
was
pl AUDI ✗ )
mutually
are
D
=
that the selected
need
we
.
produces
4 D
,
p( ✗ / B) =÷o= 0.04 P( ✗ 14=0-03 PINDI :O -02
0.05
=
C
,
%j= 0.2T
of selection of defective item
event
=
B
,
p (B) =
0.2
=
the
A
=
.
?
we
have
①
have
we
,
PIA ) PC ✗ IA )
.
=
P (A) .PH/A)tpCB).pCx1Bi-pcc7.pCX1c)tPlDtPlX/D)
0.2
✗
0.05
=
0 I
.
Or
✗
0
.
05 -1
0
.
15×0.04
+ 0.25
✗
0.03 + 0
.
Lex 0-02
07
=
Or
p( A/ ✗ I
Again
=
by
PID / ✗ )
0315
0.3175
Baye 's
-
②
Theorem
PCD )
=
we
,
-
have
PC ✗ / D)
PCAI.PH/A)+plB).pCX1B)+pCctPCX1c)-plD1.PlX/D)
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__lT-_↳÷A←
Faculty of Mathematical and Physical Sciences
-o---f__f__p___fI-o
gÉ-g _
-
PCD/ ×)
0 a
-
0.02
✗
=
PCD/
0.0315
✗
)
0.252
:
substitute ② and ③
P( AUDI ✗ )
PLAUD /
machine
by
Discrete
*
If
then it
Ex :
-
D
v1
*
it
If
is
and
is
obtain
0.252
that the ideated
or
is
D
0.575
item
manufactured
was
.
Contras random variables
random variables
a
of values
A
+
we
,
0.5715
=
probability
The
:
.
)
⑦
in
0.3175
=
✗
③
-
called
Towing
Throwing
a
or
countable
infinite number
discrete random variable
a
and
Wint
'
a
finite
takes
dice
.
and
observing
observing
the
outcome
number
the face
on
random variable takes uncountable number of values then
called non discrete or continuous random variable
a
Er :
-
.
-
it
21
Lengths
of nails
observing
the
produced by
pointer
on
the
a
machine
pedometer
@ Ramaiah University of Applied Sciences
÷-g_Es_-Bng__lT-_↳÷A←
Faculty of Mathematical and Physical Sciences
____-foII-p___fI--
ff-o-DIsxdipzbebnik.bg
-
distributions
-
for each value
It
auign
we
of a discrete random
real number plait such that
a
i)
pcni )
the function
by
p( ✗
The out of
probability
✗ = Ni
)
called
is
pen
ni
=
that
{
Ini
pcni ) )
,
92
ng
is
is
ni
a
denoted
discrete
Nn
Plan)
.
.
-
called
-
-
.
pcni ) plmo ) PIM ) Plm) Plas)
*
the values
.
.
values
M
function
probability
a
takes
✗
distribution
Mo
.
1
=
probability
the
✗
70
i :D
then
*
[
)
ii
*
plait
variable
ni
function plx ) is called
function
The distribution function fcn )
The
An
probability dimity
.
*
flag
flat
is
Mean
the
PC
=
P(
=
called
,
c-
)
✗ Eni
plno )
=
)
and
variance
variance
É
=
of dis
(
v
)
=
=
plan ) -1 Play
-1
-
-
+
pcni
)
distribution function
.
Mdi# probate distribution
of discrete
(M)
+
Épcslj )
=
cumulative
vañanu
mean
Ni
j :O
mean
The
✗
defined by
probability
xi
Teli
.
[ 4h
+
is
given by
plnil
probability
{ (ni
?
distribution
-
µ
pet
'
-
?
distribution
is
given by
pcni )
snipe)
plni )
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__lT-_↳÷A
Faculty of Mathematical and Physical Sciences
-o-_--__f__p_-→fÉ-o
ff-o-v-E.fm?pcni)-M-pcni7-2Mniplni
-
)
Enipcnitpispnii gpeznipm.IM
=
-
variance ( v )
1)
4
pl
-
l
<
finite
2)
see
N
-
plus
3
-
K
For
-
>
)
-
Tv
-
its
.
deviation
find
Also
.
pcx a- 1)
,
pcx > 1)
.
pint
sold :
r
ne
such that the
K
standard
mean
(
-
-2µL
µ
distribution
following
Hence find
probability distribution
Find the value of
a
+
ENE plni )
=
standard deviation
represents
and
'
{ nipcnil
=
*
)
I
2K
finite
K
0
I
2
3K
4K
3k
2k
probability
and
o
I
-
Epcul
=
K
distribution
I
-12k -11<=1
-13k
-12k -13k -14K
3
161<=1
1<=41670
i.
n
pm
Mean
-
'
3
116
-
I
-
I
0
4116
3/16
2116
=µ=
I
Ex poet
-
=
3
2/16
411
% ( -3-4-3+0
÷(
=
3116
2
o
-13+4+3
)
)
µ=0
variance
-
'
=v=
V
In per )
=
Eaipcx )
-
M
/ M=o
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-ng-ghf-fi.A
T
-___-faf-_p___fI-fÉ-g _
-
9×1-6
V=
%
=
+ ↳
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✗
( 9+8
-13*3+8+9
)
"
=¥;¥
V
5. D.
i
)
Ig
=
plus
¥
=
=
1)
I
=
-
pen
>
1)
iii)
pl
-
ku c-
-
%
=
pen > 1)
÷
=
pen
=
-
2)
1< ✗ c-
2)
e-
+
IT
Pln
-
3)
%
=
pln -01
=
=
pl
Pla =3 ) )
+
1-1%1
=
Ii )
-
1-1%+7,1
=
PINEY
( p( a- g)
2)
=
+
Pln
-
1)
+
Pln -4
-
÷ -1%-1 :
%
@ Ramaiah University of Applied Sciences
Faculty of Mathematical and Physical Sciences
i-ag-of-o-nge-L
T-fi.AT
____-faf-_p___fI-ff--___
-
2)
probability duuihi function
following table
II
the
K
ploy
Find
finite
3K
the
the value of K for which
distribution
probability
a
variate
is
✗
given
by
13k
11k
9K
7k
5k
of
Also
given
distribution
is
find
plnzs-1.pl
PINE 4)
32×2-61
PIM
70
Splat
4
-
49K
.
D
.
.
l
K -13K -15K -17K
=)
S
distribution
probability
finite
the
For
soy
&
mean
,
,
.
-19k +11K
-113K
=
I
I
=
1<=1149
'
PIN
it
pin >
ii )
pls
149
5) =
<
see
5/49 Flag 9/49 1114g
3149
¥a+¥a= IF
6)
÷,
=
)
plus
4)
1-
=
=
( Pln
1-
=
=
¥g -1¥,
+
3÷g
=
iii
13/49
1-
-
5)
1¥,
+
+
pcn
-61
-
)
Is )
2¥,
=
?÷
@ Ramaiah University of Applied Sciences
÷-g_ES_-Bng__h
T-f÷i&
Faculty of Mathematical and Physical Sciences
Mean
=
µ
Enpcx )
-
¥(
=
=
3 -110
+
[ a- pint
v=
tag (
=
tag
=
-
973
✗
-
,
.
Tv
=
AT
+31-20+63+144 -1275-1468 )
0
9¥
=
s.ir
-121+36-155+78 )
÷,
=
variance
0
-
4120,92
¥0,1
92¥
=
-4¥)~
-
4¥91
Berni 's theorem
the
probability
nenpnq
of
""
-
success
where
q
Moti :
a
The
is
p
the
teem
i.
e
peg
trials
Ihi
probability
is
=
n
is
equal
probability of
probability of failure
is
of
of failure
in
I
one
excess
of
to
and
.
success
&
probability
Bird distribution
If
p
of failure
,
by
pcnkncnpngn
It
1,2
-
of µ -1p/m
this
or
Itn
are
in
"
g
=
-1
"
"
c.
-
Hmu
Mean
=
pq
'
"
g
-1
ne
,
pg
"
+
.
.
.
+
v=
became
of
Binomial Distribution
"
.
-
'
"
=
"
nezp g 't
+
pen ,
is
.
called
np
[ ripen ) pet
-
standard deviation
and
pn
=
npq
n=0
*
given
.
function
In play
µ=
"
nap g
is called
"
the
=
probability
is
trials
pn
.
n
variance
the
n
2=0
☒
n
-
.
.
+
Bernoulli Distribution
=
n
-
p2qn~
ng
distribution
{ pen )
*
l
is
observed that the value of
each
pen, for
the teecceitive terms in the binomial enpanlim
be
may
.
-
q
of
out
naevus
2
nqpqh
qn
and
success
n
-
l
O
per )
of
probability
An
n
n=o ,
probabilityof
Hu
is
= - =
Tv
=
Rpg
-
-
+
p"=(q+pi=
1hrs1
probability function
.
1)
when
getting
a
Exactly
i)
one
head
s
heads
utmost
atleast
iii )
of
heads
2
exactly
in
probability
find the
,
iil
sod :
four times
toned
is
coin
heads
two
-
know that
we
given
and
Therefore
it
the
by
we
PCH
plot
pent
2
I
0
n
Pls )
Plz)
probability
of
"
pink heap g
oleo
know
of
head / =
Pln
where
exactly
1)
-
Ac
=
1- p=
"
iil
probability
plat
most
head
of
s
)
=
4
getting
heads )
=
-
I
=
=
-
1
-
is
'
(o -5110.5 }
•
lost
( 0.574
°
o.si
at most
I
=
4
pq
,
.
4-
-
]
=
pct
-
p q
,
y?÷
=
-
trials
1-0.5=05
head
one
Ac
=
=
n
n
PCH 1=1-2--0.5
failure q=
-
of
out
ueecees
a
p=
probability getting
Pli
Play
" "
that
probability
4
3
p(
u
>
-
-
4cg
p
"
heads
a)
"
p
ga
qn
-4
-4
"
is
plat
most
heads /
s
1
=
I
=
iiil
probability
getting
of
p( atleast
2
I
I
=
-
I
-
14
'
2
probability
I
-
I
heads )=
"
Coq
"
5
.
.
5)
pcu =D
+
°
he
+
"
+
co 4. 5)
( lo
-
"
-
c
"
+
is
,
P
,
pg
,÷
10.574
exactly
the heads /
c-
heads
two
is
Pln -21
-
=
" "
nczp 'q
=
"
Cz
=
6
.
exactly
2
heads)
"
(
o
*¥
=
P(
(0-5) ? 10.53£
¥?_②
=
=
•
( 0.5)
0.375
5)
.
.co.il
"
)
(0-514)
d. 5)4)
a.
+
" '
&)
,
"
)
0.6875
getting
of
plena
(
-
Pln :o)
( neo p°gh
(
-
=
p( atleast
-
heads
two
(
1-
=
4
9375
.
atleast
heads )
=
"
G. 5)
-
0
=
p
-
"
②
probability
The
defective
be
what
is
the
ii )
iii )
we
-
given
it
Yio If
is
that
2
factory
a
,
are
are
are
probability
of
the
defective
a
non
a
probability
"
p( exactly
2
-
-
2)
p
defective pen
is
out
of
where
/
Pln
=
12cg
=
=
122£
66
=
plat least
2
(
defective
/
I
=
I
-
-
(
(
2)
g-
-
n
l
-
P
trials
-
-
neap qn
9)
.
=
0.9
is
-2
'°
10.95°
? (0
9)
=
0.230
( Pln -01
-
ai 't
plan ))
+
"
-
'°
.
Keo p° q
lo
-
12
-
=
(0-1) ?
0.1 )
1-
=
-
1110--0.1
-
( o.it/o.gj0
.
defective )
2
=
.
-
n
10.1510
.
=
p( exactly
pen
ueecees
a
" "
defective
.
is
of
pink heap g
Pln
til
by
manufactured
are
pens
defective
defective
atleast 2
them
defective
none of
Exactly
of
by
inch
12
.
probability
know that
pen manufactured
a
probability
i)
80¥ !
that
-
z
°
+
.
pig )
"
12C
,
(0.1110-9)
"
)
pl atleast
2
defective /
1-
=
0.3409
=
iiil
p(
no
defective )
Pln -01
=
=
=
=
(10.9%+12.10-1110.9) )
"
-
p° qh
Nco
µ
.
12
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0.282
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correlation
analysis
and
correlation
In
variables
n
b/w
ray
number v called as correlationy
two
1) volume
E
2)
Two
of
coins
and
Types of correlation
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the
By plotting
given
for
i
i. i.
-
said
is
-
.
positive
i)
or
Negative
or
inverse
iii)
Linear
if
all the
Non-linear
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of
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iii.
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-
.
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if
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increases
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be
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a
pair of random variables lniisi )
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in
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Rainfall
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pzpert-iisot.com#
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in
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C- 1
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I
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is
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✗
suppose
and
y
equation
in
if
regression
Y
depends
✗
known
=
too
+
rumination
Take
Ey
=
Ty
of
From
①
end
q=
@
=
we
,
resign
.
re
on
y
.
Zant
+
+
as
Et
n
J
U
Earn
+
MI
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on
both sides of of ①
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n
linear
simple
big
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variables
them
regression
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two
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line
on
n
then
re
on
Clotaire
as
If
is
study
dependent
=
between
the
relationship
of
is
y
Y
is known
MY
the
Ao
n
+
as
I
-
②
get
where
EXY_
F-
E✗Éy
sa
-
✗=
a
Y
y
:
-
-
Ñ
j
FF=F÷
srT÷÷=F÷
.
For the
1)
coefficient
H
Y
sod !
fit
a
57
67
67
56
65
68
Y
65
67
66
56
line
regression
66
N
calculate Itu correlation
randy
65
n=8=
.
68
69
70
72
72
72
69
77
length / N
✗ =n
-
I
y=y -5
65--66.75=-1.75
-0.75
✗
-0.5
2
ya
✗
Y
3.0625
0.25
0.875
-11.5
0.5625
132.25
g. 625
-2.5
95.0625
6.25
24.375
57
65
-9.75
67
68
0-25
0.5
0.0625
0.25
0.125
68
72
I
-25
4.5
1.5625
20.25
5.625
69
72
2.25
4.5
5.0625
20.25
10.125
70
69
3.25
1.5
10.5625
2.25
4.875
5.35
3.5
12.25
18375
F-
Ex :
/
data
given
and
7
534
EY=
'
27.5625
2×2=143-5
540
ñ=E÷
=
a-
Correlation
co
r=
r=
=
-
-y=E÷
5,3-4
¥0
=
Ty
66.75
efficient
EY "-194 EXY -73
r
=
is
67.5
given by
Exy
¥µ=143.50194N
0.4375
→
No
correlation
-
line
Regreuion
y
a. +
:
given by
is
Ey
an
①
WKT
:rs÷n
a
③
-
sy=✓¥
sy=F÷
=
Sn
③
aotarñ
=
sn=✓
where
-
T
g-
ay
Sy
4.5276
=
=
5.2644
5.264€
0.437g ,
=
4. 5276
An
put
0-5086
=
9--0.5086
Ñ
=
Ao -1
Ao
-
Ao
Y
=
O
-
67.5
-
5086
-
O
-
Ñ
5086
✗
33.5509
aosq
=
Ñ
0.5086
J
-
=
substitute
eef ②
in
in
33.5509
1-
equation
0.5086N
/
Sao -1922
T
T
-
①
,
66.75
②
Ear
.
now
obtain the
2)
regression
a
goin !
✗
I
3
4
5
Y
2
6
8
10
-
n
a
En
y
7
14
✗ in
-
ñ
yay -5
✗
~
y
'
✗
y
-3.66
-7-33
13-395
53.728
26-827
-1.66
-3.33
2.755
11.088
5.527
0.4352
1.
6
4
8
5
10
0.34
0.67
0.1156
7
14
2.34
4.67
5-
8
16
3.34
6.67
-
0.66
1.33
-
475
11.155
0.877
768
0.448
0.227
21.808
10.927
44.488
22-277
{ ✗ 2=33.330 EYE 133.3285×4=66.662
-56
-
a-
End
=
g-
co
-
g-
efficient
r=E✗Y_
yare
9.33
=
r
is
given by
66.662
=
⇐y
and
%
=
Ñ= 4.66
Correlation
End
=
7-
=
a
16
-6
3
i.
fit
-
2
EY
and
8
l
-28
-
correlation coefficient b/w
nosy
line
-
-
lÉ,y=
positively
correlated
0
-99g
line
Regression
y
ao
:
given by
is
-1
Ey
an
-
=
g-
a-
I
I
①
Eao -19 En
=
ñ
an + as
-
②
WKT
an
where
:r¥n
③
-
sn=F
ss=F÷
=F¥
=F÷
Sn
2
:
Sy
5818
.
5-1638
=
③
q
r
:
Ay
put
ao
Ñ
=
Ao
Y
The
ao
=
=
5j!%÷
②
Ñ
9998
-
9. 33
=
Go
substitute
1.
✗
9998
eg1
in
Aot
=
0.999
=
1-
=
q= 1.9998
4-
Sign
.
1.9998
-
I
-
Ñ
9998
✗
4. 66
0.0109
ear
in
est
①
0.0109 -1 1.9998k
required
regression
line that
fits the given data
.