Maths uestion Ban

Unit-2
Find the Comple mentaiy fun ction of CD3+1 y COSax
(b+1)y Cvsa

So y- CF
auxilay equation 15 Fm):u

m+ o

C a t b3 = (atb) lat-abt b)

(mt) (m-m+) =b
m -1 m-mtl=0

m 1t1-4

m:13 . Ni

CF CE"+ (C,tos C3 Sin x) e

a Solve b-3D°-3D+4D+4y=o

D-D -3D2+u Dt 4)y=O

auzilany equation is flm)=o

Cm- am-3 m +4m+4)y =o

m am' - 3m +sm +4 =0

:-1 1S bne YOot

|I -3

m O 1 3 -
-3 O
m-3m+ 4 M =O

m: - IS anbthey YOot
 3 0 4
m- -14- 4

m- ym t4 0 (m-2)- o
Cm-a) tm-3) o
m ,

m-1, -1, 2,2

Solution: y>CF
y (Ci+C a)e* + CC3t Cua)e*

3 Solve lD3- 5D+8D-4) y-b

(D- 5D+ 8 D- 4)y=o
auailary equation is Plm) =o
m 5m+8m-4 =
m= is one TOOt

5 8 - L

m
4
-4 4
m-4m +4 o - m-a)'-o
m-2) (m-2) = d

m , 2,3

Solution: y: CF

y C1e*+ CC2t C3) e2

( D + 2z^D+2)y =o

(3p3 a2' D'+a)y0-0

et ea 0 is in the foim DP Cauchys Euler diffevential

equat10n

tet 2D =6 , D°- el0-1), *3p- ole-1) l -)

 wheve e=> e-> tog = 2

o(o-1)te-a) + 20 (0-1) +2)y =

(e o3042) + 20- 20+2)y=6
(o3-30'+26 + 2o6+ 2)y=o
(3-0+a)y-0
auxilay equation is f(m)-o

m-m+3 = O
m-I is one oot

|1- 1 D
m--1 -1:
2 2 o

m- a m +3=D

m 2 t J4- 4l)t2)

m a tJ-4 1i

m:1, t i

Solution y= CP
y Ce+ (C, Co5 2+ C38inz) ez
CC2 Cos (Loga) + C3 Sirnt toga)*
5 Find the complementa9 Functi on df D+1Dy= osan
Refer No
Solve2Dt3 Sin
Sin
D43
D-3
Sin
(D+3) (D-3)

D-3 - Sinx

D-9
 put D
D3sina D(sinx)
-1-9 -D

DCOSX- 3sin)

Find the solution of the dif Pevential equati on tD'+4)y=o

Qunilay equation is fim)=b

m -4
m t-4
m: tai

Soluti on y= CCF

y CC Cos aGSinax)e
8 Solve (a*D* - 4 D + 6)y=0

(xD-yzD4C)y=o-0
ea o i s in the foim of Cauthy's Euley dlifPeventi al
equation

et D e, D?= ©lo-1)

where e=> Log 2

(ete-1)- 40+6)y=o
(e2--496) y=0
e2-50+6)y- o
auilary equation is flm)=0

m-SmtG D

m-am-3mt G op
mm-2)-3( m-a)=o

Cm-3) (m-3)
m 2, 3
 Solution y: CF

CD45Dt c)y = e
Solve
D 4 SD +6) y = e

flm)=b
auxilary equation is
m t 5 m tG = D

m+am+3m + 6 =0
mtmta) +3(m-+3)=0

Cm+a) (m+3)=O
m: 2 , -3

CF = CIe Ca

PI e
D4 5D+4G
put D

e
1+5tG

ID Solve d-=0
A-20
dt3

iven d =0 -

0
d3

eq O in operatoY fom is

(p3-1) O
auxilay equation is fm)=O

m- o a3b3 =
(0-b) (a?4 abt b'))
(m-1) m*+m+1) = D

m mtmt1 =O
- ItN1-4 t3

m-,1ENei
Solution y C 72

Cie+ (C, (os a t Czsin 3z

)e
I1 Find particulay int cqral Df (D6- pu) = **

(D-D4 ) x2 -D

PI .

D-D4

-D+D6 -D+ (1-D°)

1-DSa
D+

(1-D I+ D+D'4.
-

+D) *2
D4
L'+a)
D4

(7*)
()
PART-B
Solue - a t39 =e"tan by method of waiation
dz2
PaYameters.

d 2 + y e tan -0
d
P2 Q 2 R= etan
form is
ea Din opeiatoDY
(D 2D+ 2) y e tan -
1S ftm )= O
Auilay equation
m-am + =D

m- u-B

m tN-4 1ti

CF =(Ci COS
+ C Sin ) e

CF Aua) +8v(1) Ciuc)t C V(X)

ula)= CoS eX V(x) Sinxc*
u'(a) COSX e+-eX sin
e(CoS -Sinx)

V'a) = Sinz e" + e coSX

=e COSx tsina)
2
uv'-vu' =e ? CoSX( Coszt Sinx)-e Sin Cosx- sin)

= ea(cos'1+ (oSz6in
-

Cosx sAnz 4sin'a)

l T PI AutX)t Bv(x)

sin eM. tan-

A-VR
d dz

Juv'-vul

sin'2 d CoS d
CoS
CoS COS

-CseCz- COS ) dx
Logl se ( a t tana]-sìn
sin- Jog] Se ca4 tan2

B uR Ce tosz 2 tan
d
Uv'-vu e

sind di - (DS :

Au(a) CoS2 e^ Sinz - tOS e , Logl Set t tana

Bvx) -

e tosx4S i n
Solution y: CF P I
cog|secxttanx|
e Cos sihx (oS
-

y (CicoSx + (, Sinx)et
+

- eYtoy'a Sifx

(C COSx C2sinx) e CoS ae. Jog| Secttana

a. Solve CD4-1)y =e" cos

D-1) y el cosx
cquation is f(m)=O
auxilay

C a-b- (atb)(a-b))
(m41) m-1) - 0

m-ti

CF (Ci CoS + ( Sinx) + C3 e+ Cue

P:I e ws
D4-1

put D= Dt

e -oS
(D+I

e CoS
D 4D3+ 6DH4 DM

e - COSX

b 4D + GD+ 4D
put D' -I

e COSX
-+u-1)-64 46
e COS
1-G
,
CDS

Solution y: (Ci
CoSX 4 (Sina) + C3e4 Ce*- osz
 by the method dF vaTiatiDn of
3 Solve CD41Dy COSeCxx
Paramete1s
CD41Dy COSeC -0

Auxilay cquation is f(m)-b

m tN

CE C CoS X + (> sinx

CF C u(a) t C> V(A)

uta = COS vUa)= Sinx

u'(a -Sinx v'()= CoS

uv-vu = CoSx+ Sin

lbT PI A u (a) t BV(A)

A -VR Sinx CDSec

-d
Juv-vul

-sirh dz
Sip
-X

uR
dd cosx Cosecz d
uv'vu

(CoS d
Sin
tDgIsinx |
Solution y CF +PL

U: C CoS7 t C2 Sind - LoS+ log1Sinx |Sin

4Solve (D2-y0tu)y 8x esina
Giiven (DuD 4)y-8x csinax -DD
auxilay cquation is f (m) =
O

m - u m +u D

(m-2)- 0

(m-a) tm-2)-0
m- ,

CFCC t (,%)c?

PI 8e2 ? sin a
D-y D+u

put D D+2

82 Sin 2
CD+2) y(D+2)+

8e2 Sin 2x

8e27 - Sin 2
D2

isina2
e2 COs27+
is the I P of el2
Sin 32

8e IP of ei3
put D Dt3i

8e37 P of ei2
D+2i

8e IP of elD_
D1 4Di-4

8e
9 IP of eld *
y4Di4D
 - 8ee IP of el3x
el 1
-8
ul-(Di+ R)
-a P of ei2(1-(Di+
C1-D l+D + D+ -

2c 1P of e13 (14 CDit ) + (Di+ ) ) x

- 22 IP of ei32 ( + ai+-2)
- e IP of e ( (3)+2%i)
-a2 TP of (tos 32 t isinaa( a2-3)+ 2ai)
-
2 e27(a2cos32 t Sinax (**-
Solution: y= CF +PI

y C C t C2x ) e - a2 (3 Cos ax + 5inaz (*-))

5 Solve D-5Dt 6) y= e *

iven CD2-5Dt 6)y- *e -0

auxilay equation is ftm) D

m-5m +6 =0
m-2 -3m+ 6= D

m (m-)-3(m-2)=0
m-3) (m-3)= O

m- , 3

CF C e?x 4 Cze3*

P.T D5D G -

Cput D = D+1)

e
(D+I-50+4)+6
eX
D++D-5D-5+6
D+I42D-5p++
e
D3D + 2
 3D+ D?

2(1-3D+D)

(1-3D4D)-
-D'= 1+D+ D4

1-3D D )
-ap )
-34to)

Solution y= CF+PI

6 Solve the differential equation (D-5Dt 6)y eSina

D-5D+ 6)y e sin - D

Auilary equation is flm)-o

m-5m + 6 0
m- am -3m+ 6=D

mm-2)-3(m-)=0
Cm-a) (m-3)=o
m: 2, 3

CF Ce 27 + C, e3
PT e sin
D-5D
put D= Dt

e Sin
CD+1-5lD+1D+6
Sio
D4142D -5D-St6
 Put D 1
Sin
e
-14-3D+1
Sin
3D

e 3D _Sin
1-902

Put D-

(143D) Sina
ex (Sin +3 (oSa)

T0
SOlution :y-(F+ PI
Cie+Ca¬*4 sinz43 COS
=*e°-Sin243
(D2Dt)y
diffeventiai cquation
SOve the

Gilven (D-a0+)y-0
is flm)=0
AulilaYy equation

m2-am+)y-0
m2-am+1=0

(m-1)- o
m-1)(m -1)=0

m ,
CF CI C22) e^
( )
P.l Put D D+3
-2D+1
3
2

CD+3-a (D+3)+1

e
3 1 2 Q2

D4946D- aD- 6tl

3X

4 +4D4 D2
 3

4(14 D+)
(14D) 1-D+D".
3*

(1D-p)x
4
3X

4 (- +a) (2+)
PI - Sinox
D-2D+1
put D -

Sina sin2S
-44D+J -D-3

2D-3)
4D-9
sina
put D- -4

( D-3) sinox)
95

(
5
3D-3) Sin 2x 4 CDS 37 -3sin 9 )

P I3 3e
D Dt1 put D O

3 e

3
Solution y: CF+ PI1t PI, + PI3

CC+ Cx)e^+ e ( 7+ 3 ) - (y tosax 3sin3*) +3

y
L
 diffevential equation
8 Solve the

(x3D 3x'D+a)y 1o[

1ox
i v en (*D4 3aa'D'+ ay
aiffevential equation
Cauthys tulcY
WB:T eq D is
D(O-1) 3Dol0-1) t0-a)
tet D O D
e
whee

oe-1)(0-2)+ 2olD-1)+ 3)y 1Det loe

ote-30+ 9) + 20- 394 3 )y loe tIDe

(33e461 3e?-a6+a)y= toe+1oëz
te3e'+ 2)y- 10e+ tDeZ
auxilay equation is flm) D

m-1 is one oot

m-D 2

m-am+ a =0

m3tNu-41)3) 9t -s

m: -I, 11

CF C e t CC2 Cos 2 + C2 Sin2)e

PI IDe
+2

put 1

e2 Se2
A+2
PI De

put -I

1-1

Case of failure

t02 2
302- 20
Put e -1

I02
32 102 5

Sotution y CF+PI+ PI
: Cie+ CC CDS24 C3 sin2) e+ 5e'+ 92ë

+ CCa CoS Cogu)+ C3Sin (log1) *+ 5 + 2 Jog2

9 Solved,- 3 + 2y 37
aetSin 2x
3
Given dy a dy 4 ay e°*+Sin 3aO
dr
e D in operoutDY fom is

CD 3D42)y e?*+ Sin 2

auxila1y equation is flm)= o

(m2m 49) y=0

m-3m+ 3 =0
m-9mn -m+3=O

m (m-) -1 (m-)=D
(m-3) m-1) =u

CF Ce + C, e2
0 SOlve (142) d'y + ( t ) + 9 2sin (10q (4))
d
Gnven (14 d'y + H) dY +y= 2Sin ( loq tHa)) ->0
da2 da

in opevato fovm is
eq.o

(1470 D+ (47)Dt Dy- 2Sin

(1og lI42))->
.K.T eq-2 is tleqencse's equation

let14a)D = e (I402= e (B-1)

(42) = z
where 1+ e lq
2Sinz
(e (e-1)+ 0+1)y -

-++)y= 2Sinz

(+)y 2Sinz

f(m) = 0
auxilary Equation is
m + l = OD

m-1
m= ti

CF C COs z + C2. Sin z

2sinz
P-I=
e2+1
Pt e
2 Sinz
-1+1
Case of faüluve

2sin z
zOS Z

SOIution: y= CF+ PI
C2Sin z - z o sz
-CCCSZ t

y= C0S tLogt+) + Ca Sin (Dgi+) - Log C1+2) Cos(Log tl47))

1-Solve CD-4D +3 y Sin3X cos3
D- 4p+ 3) y sin3a Cos ax -0

auila7y equation is +m)=0

m-3m - mt3-O
mtm-3)-1( m-3)=O
(m-3) tm-1)= d
m 1,3
CFCie%+ 3

P-I s i n 3 CoS ax
D p+3
sin3xa (Sin3x cos ax)
D- y D+3

C sinACo$B = Sin (At B) Sin lA-B))

D24D+3
Sinsz +sin )
D 4D+3
P I Sin5
D4D+3
put D -

25

- 25-4D+3
Sin5 ) SinSx
-4 D-22

aD- sin5
4D-121
put D- -25

- aD- sins (s Io CosS- |sinsa)

-ID0-12 884

a 1OoS5x- 11sin5)
884
sin
PT 2 D4P43
put D -I
Sin Sin
-1 -s-Dt4 3 -4D+2
 Sinx
21-aD
142D SInx
-4D
Put D= -J
H 3 D J4aD sin S ina t 2 CoSa)
2
14 4
Solution y= CF+ PI +PI2
y Cie+ C e3%+ 88 4
(toCosSX - 11sin5a)t sinx t 200Sx)