02Aug
ust 2022
ay- ues day
inSt Offune lectusne (a
AR11AL AEFEREN1LAL 111 KgP
2 Yea1 (30 SenMeS+e1)
QUATIONS
Lectue 01 1Vt1Ooluct/on to PDE
8:00AM- 10:00AM
PDE:quotHOn patHal denivatves wnt Vmone
Which involve3
han one inolependent vas1iab/e 8 colled a panHal diklenol
equawon (PDE)
*091ole91 and Segmee of a eeMevtiak tquatton:
Q91dlesAue O91ole 0f a difestevtial equation is e
(onde
powen
of Hhe highest olesnivative involvedin te dikkenental equatian.
eggee: Ahe dege1ee of a dikfenetal equahon 8 the powe
Of he higlhest o91de1 des1ivative involved in Hhe equation w e n
as an as
the equatHOn_ ha8 been made 1aHOnal and iMtego1al
he demivatves_aMe_1cOcegnnegl.
quato C1den degiee
d y KX 2
dy2
(du /dor): 4(d4/dox)-2y 1
(ovlat) K (V/a) 3 2
y nola) c (dnlay) 1 2
(1+ d'9/dn°2 - a d'y/dw2 3
Paperkr.att
, 02 AuguS+ 2022
oy Muesdoy
F(%«» 12, Kz. Kn , Z, 2n1 , Z x , . . Znn): O a PD.
lg-002
9 8 independtnt vaonabte and z i dependent varable.
dz/8M1
9epivatHon of a PDE: (Lectune 02)
Metnod By euiminatom 0f abitraMy constoants:(&yane inde
f(n y, E,a, b) = o - (1)
and t depenolent vanable)
P: dz/d 7P
Sierentbtig paotwally wnt : of+df(0z O-(2)
SenetHating paoialty wnt y: d f f d z = 0 )
dy
EliminaHing )and (6) fonom 4),(2) and (3), we have
f(y, Z P:9)= o 18tandev1 PD¬
Pp+@qRStandond foonnm of 18omdem P.D-E
P,R 01e funcius of K«y, z p- d7/ld 9 0z/dy
Method By euimin0ton of_a1bite1a9ty funchons
Let u and v be tme uncHons af , y aud z whith
-(t)
COMmeCted by f(u,v) :
=
oikkenenHating patiailty wot and y,
df Ov+dv. oz -(2)
u 0
0(du + ou. dz+ dfdv+ -(3)
ou dy oy ov dy
aerkraft
, ay- uesday 02AuguNS+ 2022
tuinninohing df and df uonn (2) aud (), d-003
dv
du + du P auou. q
dy dz
dy
dv. u dv. du pP du d -du dv
dz
dy 3z dy dz
9 0 18 omden PDE
Pp +
EMample 01 om a PDE by eunminaHng (a,b) fort z:(+a)(y+b
3z - y+ b) P 18atep: ikerrentiate pa1ialuy w t
nd
Oz ( + a): 9 2"step: 0ktesnevttiate_paottalky Wt y
dy
Z =P (Requined PDE)
fonom +y+z1
EHaMple 02 dosmm a PDE by elininating a bc b c
aln pKeenevtaing pontiauy wt a n d y:
2 2 c + a'z d?/8N =0 -(2)
a2
8imilasy C'y + b'z d7/8y =O - )
C2
iorentiaing (2) paHaly W:91t , We get c a - Z d z "
a(0z also,
a
2
-Z dz + zo +dz)= 0
9 d
gimikorny dy dy dy
iakrat
ust 2022
ay- ues day
inSt Offune lectusne (a
AR11AL AEFEREN1LAL 111 KgP
2 Yea1 (30 SenMeS+e1)
QUATIONS
Lectue 01 1Vt1Ooluct/on to PDE
8:00AM- 10:00AM
PDE:quotHOn patHal denivatves wnt Vmone
Which involve3
han one inolependent vas1iab/e 8 colled a panHal diklenol
equawon (PDE)
*091ole91 and Segmee of a eeMevtiak tquatton:
Q91dlesAue O91ole 0f a difestevtial equation is e
(onde
powen
of Hhe highest olesnivative involvedin te dikkenental equatian.
eggee: Ahe dege1ee of a dikfenetal equahon 8 the powe
Of he higlhest o91de1 des1ivative involved in Hhe equation w e n
as an as
the equatHOn_ ha8 been made 1aHOnal and iMtego1al
he demivatves_aMe_1cOcegnnegl.
quato C1den degiee
d y KX 2
dy2
(du /dor): 4(d4/dox)-2y 1
(ovlat) K (V/a) 3 2
y nola) c (dnlay) 1 2
(1+ d'9/dn°2 - a d'y/dw2 3
Paperkr.att
, 02 AuguS+ 2022
oy Muesdoy
F(%«» 12, Kz. Kn , Z, 2n1 , Z x , . . Znn): O a PD.
lg-002
9 8 independtnt vaonabte and z i dependent varable.
dz/8M1
9epivatHon of a PDE: (Lectune 02)
Metnod By euiminatom 0f abitraMy constoants:(&yane inde
f(n y, E,a, b) = o - (1)
and t depenolent vanable)
P: dz/d 7P
Sierentbtig paotwally wnt : of+df(0z O-(2)
SenetHating paoialty wnt y: d f f d z = 0 )
dy
EliminaHing )and (6) fonom 4),(2) and (3), we have
f(y, Z P:9)= o 18tandev1 PD¬
Pp+@qRStandond foonnm of 18omdem P.D-E
P,R 01e funcius of K«y, z p- d7/ld 9 0z/dy
Method By euimin0ton of_a1bite1a9ty funchons
Let u and v be tme uncHons af , y aud z whith
-(t)
COMmeCted by f(u,v) :
=
oikkenenHating patiailty wot and y,
df Ov+dv. oz -(2)
u 0
0(du + ou. dz+ dfdv+ -(3)
ou dy oy ov dy
aerkraft
, ay- uesday 02AuguNS+ 2022
tuinninohing df and df uonn (2) aud (), d-003
dv
du + du P auou. q
dy dz
dy
dv. u dv. du pP du d -du dv
dz
dy 3z dy dz
9 0 18 omden PDE
Pp +
EMample 01 om a PDE by eunminaHng (a,b) fort z:(+a)(y+b
3z - y+ b) P 18atep: ikerrentiate pa1ialuy w t
nd
Oz ( + a): 9 2"step: 0ktesnevttiate_paottalky Wt y
dy
Z =P (Requined PDE)
fonom +y+z1
EHaMple 02 dosmm a PDE by elininating a bc b c
aln pKeenevtaing pontiauy wt a n d y:
2 2 c + a'z d?/8N =0 -(2)
a2
8imilasy C'y + b'z d7/8y =O - )
C2
iorentiaing (2) paHaly W:91t , We get c a - Z d z "
a(0z also,
a
2
-Z dz + zo +dz)= 0
9 d
gimikorny dy dy dy
iakrat
