IAPPENDIXI

MULTIPLE CHOlCE UESnONS MCi ·s

Ch, 1 Linear Differential E ations with Constant Coefficient
Marks
~ ..a.mental')' Functions :
1YP' 1-c~··
· m m , ... , mn of aux,·1·iary equation
2 3
. «D)=0 are real and distinct, then solution of q,(D) y =0 is .......... .. (1)
he ro0ts "11, ,
1, Jft 1111X+ clT12X + ''' + Cne mx
n (8) C1 cos m1X + C2 cos m2X + ''' + Cn cos mnX
(Al c1e
c1X + m2ec2x + ··· + mnCnX (D) C1 Sin · m1X + C2 Sin · m2X + ,,, + Cn Sin
· mnX
(Cl niie m m- , m .. . , mn of aux -- 11
Tary-equation
- - · q,(D) = 0 are real. If two of these roots are repeated say m1 = m2 and th e
2 3
. The r-~~s r;ts m , m4, .. ., mn are distinct then solution of q,(D) y = o is ............ (1)
2 3
rerna1n 9
m1X + c mix + " · + CnmnX (8) (C1X + C2) COS m1X + C3 COS m~ + X ... + Cn COS mnX
2
(A) c1e
(ciX + c ) em1x + C3 em3x + " · + cnemnX (D)
· (c1x + c2) sin m1x + c3sin m3x + ... + Cn sin · mnX
2
(C) f
°
·n ..... , ·
?
The roots m , m2, m3 ... , mn aux,.._,. ~uat,on q,(D) = are real. If three of these roots are repeated, say, m1 = m2 -- ffi3 and
1
3. the rema""· ••:i roots m4, ms, .. . mn are- distinct then solution of ""
",1n11 t1J.D) y-= o ·1s .... . ..... .. (1)
(A) c1em1x + C2m2x + .. . + Cn emnX (B) (c1x2 + C2>< + C3} em1x + '4em4x + ... + Cne-m"x
(O (cl + c-iJ, + cJ cos m1X + '4 cos m,ve + ... + c0 cos m0 x (D) (cl + c2x + c3) sin mi>< + '4 sin m4x + ... + Cn sin mnX
f m = a + i~ and m2 = a - i~ are two complex roots of auxiliary equation of second order DE 4>(D) y = 0 then it's solution is
t I 1 W
............
(A) P!- [c
cos we + C2 sin ax] (B} eax [(c1x + c2} cos ~x + (c3x + '4} sin ~x]
1
(C) c eax + clx (D) eax [c1 cos ~x + c2 sin ~xl
1
=
If the complex roots m1 a + i~ and m2 =a - i~ of auxiliary equation of fourth order DE 41(D) y =Oare repeated twice then
s.
it's solution is ............ (1}
(A) ~ [c cos ax .., C2 sin ax] (8} ecrx [(c 1x + c2} cos ~x + (c3x + '4} sin ~x]
1
(C) (c x + ci) eax + (c 3x + '4) e~x (D) eax lc1 cos ~x + c2 sin ~x]
1
6. The solution of differential equation~- S Tx + 6y :;; 0 is ............ (1)

7. The solution of differential equation ~:~ - 5 ~- 6y = 0 is ............ (1)

3 3
(A) C1e-x + C2e6x (8) C1e-ix + C2e- x (() C1e x + C2e2x

8, The solution of differential equation 2 ~ - ~ - lOy = 0 is .......... .. (1)

.L
(B) C1e-2x + C2e- 2

lx
ix -2x 2
(D) C1e + C2e
(C) c1e-2x + c2e2
d2
9· The solution of differential equation~- 4y = 0 is ........... . (1)

(A) (C1X + C2) e2x

(D) Cle2x + Cie-2x
(C) C1 cos 2x + c sin 2x
2
2
d ski · (1)
lO. The solution of differential equation~- dx - 2Y = Ois .......... ..

(A.l)

I
 APPINDtX : MULTIPLI CHOICE QUISTJoHs
MATHEMATICS - m(Mech. Engg. Group) (5-D) (A-2)
ENGINEERrNG
~ . . ~gy . (1)
_ The solution of differential equation 2 dx2 - 5 dx + 3y = o is ... ....... ..
11
l ,
2 (8) 2x -3x
(A) c1e' + cie C1e + c.,e

f d.ff . I . ~ ~
equation dx2 + 2 dx + y = O is ........... .
(1)
12. The solution o I erentIa
(A) C1e 2' + C2e· (B) C1e• + C2e-• (C) (C1X + C2) e-1!

~ + y = Ois ........ .. ..
(l )
13, The solution of differential equation 4 ~ - 4

!
2
(0 C1 cos 2x + C2 sin 2x (D) (C1 + C2X) e

14. The solution of differential equation~- 4 f + 4y = 0 is ........... .

(1)

4
(A) (C1X + C2) e-2•
2
(8) (C1X + C2) e- x (C) C18 ' + C2e-4•

15. The solution of differential equation~ + 6 ~ + 9y = 0 is ........... . (1)

3 3 2
(A) C1e- 6• + c2e-9' (8) (Ci)(+ C2) e-3• (C) (C1X + C2) e ' (D) C1e • + C2e •

16. The solution of differential equation ~ + y = 0 is .......... .. (1)

(A) C1e• + C2e-• (8) (C1X + C2) e-• (C) C1 cos X + C2 sin X (D) e' (C1 cos X + C2 sin X)

17. The solution of differential equation dx2

ix + 9y = 0 is ............ (1)

(A) C1 COS 2x + C2 sin 2x (8)

3
(C1X + C2) e- • (C) c1e3x + c2e-3x (D) C1 cos 3x + c2 sin 3x

18; The solution of differential equation~ + 6 ~ + lOy = Ois ........... . (1)

(A) e-3• (c 1 cos x + c2 sin x) (B) e• (c1 cos 3x + c2 sin 3x)

5 2
(C) C1e ' + C2e ' {D) e• (c1 cos x + c2 sin x)
. 2

19. The solution of differential equation~ + ~ + y = o is ............ (1)

(A) e• (c1 cos x + c2sin x)

1X

(C) .-, [ c, cos(¥) sin(¥),] x + c2

20. The solution of differential equation 4 fy 4 2l'.

dx2 + dx + Sy = 0 is ............ (1)
(A) e-• (c1 cos 2x + c2 sin 2x)
(C) 2, (B) -x;2
e- (c1 cos x + c2sin x) e [c1 cos x + c2 sin x]

21. The solution of differential equation d f& ii'.

(D) Cie-4• + C2e-sx
• dx3 + 6 dx2 + 11 ~ + 6 - .
(A) c1e + c,e2• + c e1- dx Y - 0 1s .......... .. (2)
3
(C) cl e ' + C2e
- 2,
+ C3e -l '
 _ ID (Meeh- Engg. Group) (S-D) (A.3} APPENDIX: MULTIPLE CHOICE QUESTIONS
~THEMATICS
EERIN
6
. ~ .QY _ .
~ . I equation d 3- 7 d - 6y - 0 is ............ (2)
·on of different1a x x
ThesolLJtl 3• (B) C1e-• + C2e~2x + C336x
'
22, ' + c2e1> + c3e
(A) c1e i, c e'
_, + c2e + 3 .J3. . d2.. d
~~ .. ~ ~ ~- . (2)
·tt rential equation d 3 + 2 d 2 + d - 0 1s .......... ..
tion of dI e x x x
The souI
l~ C) (8) C1 + e-• (C2X + c3)
e' (C2X + 3
(A) Ci -1- (D) C1 + C2e• + C3e-•
·• (C X -1- C3)
(Cl e 2 • ~ ~ gy . (2)
. of differential equation dx3 - 5 dx2 + 8 dx - 4y = 0 1s
The solution
2• (B) c1e• + c2e2• +c3e '
3
- ' + (C2X + C3) e
(A) cie 2• (D) C1e-• + (C2X + C3} e- 2•
(() (C2X -1- C;) e 3
· · u 42Y o·
ZS. The solution of differential equation dx3 - dx = 1s .......... ..
(2)

2x -2x (B) c1 + c2cos 2x + c3sin 2x

(A) c1e + c2e 2 2
-2• -3• (D) C1 + C2e ' + C3e~ X
(O cie' + c2e + C3e
3
(2)
. The solution of differential equation~+ y = 0 is .......... ..
26

( ii 1
(A) C1e' + e C2cos 2 X + C3 Sin 2 X
.11)
l
(C) C1e· + e
½• (C2cos ii
2 X+ C3
' :Jj_2 )
Sin X
3
d d
dx3 + 3 ~
(2)
. of d'ff
27, The solution , erentIa . U
. I equation dx = 0 .Is ........... .

(A) Ci + C2cos X + C3 sin X (B) C1 + C2 cos Y3x + C3 sin -y3x

(C) c1 + cze{ix + c 3e- ✓3x (D) C1 cos x + c2 sin x

28. The solution of differential equation~ + ~ - 2 ~ + 12y =- 0 is (2)

(A) c1e-31 + e' {c2cos .../3x + c3 sin "3x) (8) c1e~3x + (c 2cos 3x + c3sin 3x)

(C) C1e31+ e-x {c2cos -{3x + c3 sin -{3x) (D) C1e-x + C2e--f3x + C3e-[3x

2 d
29. The solution of differential equation (D 3 - D + 3D + 5) y = 0 where D = dx is .......... .. (2)
(A) C1e·• + e• (c2cos 2x + c3 sin 2x) (B) C1e-x + (c2 cos 3x + c3sin 3x)
2 3
(() C1e-1+ e-• (C2 COS 2x + C3 sin 2x) (0) C1e--x + C2e-~ x + c3e-- •

30,The soI· ·
utlOfl of d1ffe-re-nt1ai · ~
· equation dx3 - ~
dx2 + 4 .QY
dx -4y - -o·1s ........... . (2)
(A) (C1 + C2X) e~ 2• + c3e-x (B) c1e' + C2 cos 4x + c3sin 4x
1
(() C1e + C2 cos 2x + c3 sin 2x
31. The solutio n of d'ff . . ~
I erent1al equation dx4 - y =
0 .Is .......... ..
(2)
(A) (C1X + C2) e-• + C3 cos X + '4 sin X (B) (C1X + C2) cos X + (C3X + C4) sin X
(C) (C1+ C2X + C3X 2 + c.ix 3) ex (D). . C1ex + C2e-x + C3 cos X + C4 sin X

32. The sol . . 4 2 d

ution of differential equation (D + 2D + 1) y = 0 where D = dx is ........... .
(2)
(A) (CiX+ C2) e• + (C3X + c.i) e-• (B) (C1X + C2) cos X + (C3X + C4) sin X
(C) C1e' + - x
C2e + C3 cos X + C4 sin X (D) (C1X + C2) cos 2x + (C3X + C4) sin 2x
INGINUu.G MATHIMATICS - m (Mech. Engg. Group} (5-D) (A.A} APPENDIX: MULTIPLE CHOICE QUISTJONs

33. The solution of differential equation (02 + 9) 2 y = 0, where D = d~ is ............ (2)

(A) (C1X + C2) e 3" + (C3X + '4) e- 3• (B) (C1X + C2) cos 3x + (C3X + '4) sin 3x
(C) (C1X + C2) cos 9x + (C3X + '4) sif'I 9x (D) (C1X + C2) cos X + (C3X + C4 ) sin X

34. The solution of differential equation~ + 8 ~ + 16y = 0 is ...... ..... . (2}

2
(A) c1e • + c2e-• + c3e" + c.e- 2•
) -2x
(B) (c 1x + C2) e2x + (C3X + '4 e
(C) (C1X + C2) cos 4x + (C]X + '4) sin 4x (D) (C1X + C2) cos 2x + (C3X + C4) sin 2x

35. The solution of differential equation~ + 6

dx
~
dx
+ 9 ~ = O is
dx
(2)
(A) C1X + C2 + (C]X + '4) cos .../3x + (C3X + c6) sin .../3x (B) C1X + C2 + (C3X + '4) cos 3x + (CsX + C(;) sin 3x

(() (C1X + C2) cos "'3x + (C3X + '4) sin .../3x (D) C1X + C2 + {C3X + c..) e "3x.
[ANl.mB§L
1. (A) 2. (C) 3. CB) 4. (D) 5. (B) 6. (D) 7. (A) 8. (C)
9. (D) 10. (B) 11. (A) 12. (C) 13. (D) 14. (A) 15. (B) 16. (C)
17. (D) 18. (A) 19. (C) 20. (B) 21. (C) 22. (D) 23. (B) 24. (A)
25. (D) 26. (C) 27. (B) 28. (A) 29. (A) 30. (C) 31. (D) 32. (B)
33. (B) 34. (D) 35. (A)
Type D : Particular Integral
Mark$
1. Particular Integral of linear differential equation with constant coefficient cp(D) y = f(x) is given by ........... .
(1)
1
(A) - - f(x)
+(D)
(B) l .J... _l_
+(D) f(x) (C) cp(D) f(x) (D) cp{D2) f(x)
1 d
2. ~ f(x), where D • dx and m is constant, is equal to .......... ..
(1)
(A) e""' f e-'"" dx
(C) em• f e--m• f(x) dx (D) e-mx f e= f(x) dx
1 d
3. ~ f(x), where D • dx and m is constant, is equal to ........... .
(1)
(A) e-m• f em• dx
(B) f em• f(x) dx
(C) em• f e-mx f(x) dx
(D) e-mx f em• f(x) dx
4p·11n l n d
• art1cu ar tegral cp(D) e , where D • dx and cj>(a) -,:. 0 is

(1)
(A) cl>(~ a) en (B) X _1_ ea• 1
cj>(a) (C) cp{a2) eax 1
(D) ~ eax
1
S. Particular Integral - ( D, eax where O ..Q_.
- a) • dx 1s ........... .
1
(A) -ea. x' (1)
r! (B) -eax x'
r (C) -eax
1 r! (D) x' eax
6. Particular Integral M02 sin (ax + b) h d
"" ) , w ere D s - and cp( 2) .
1 dx - a :;m 0 IS ......... . ..
(A) ~(- a2) cos (ax + b)
1 (1)
1 (B) ~ sin (ax + b)
(C) x ~ sin (ax+ b)
1
(D) ~ sin (ax + b)
 (A.5} Al'PENOIX : .MULTIPLE CHOICE QUES I IONS
ENGINEERING MATHEMATICS - m (Medi . lngg. Group
) {S-11)

1.
1 d
Particular Int-egral 4>(D2) stn (-ax+ b), wher-e D = dx and
''-
·41(-- a ) = 0, ·4> \- a
2 2)
* o· 1s· •··········· (1)

1 . ( b
l (B) x q,'(- a.2) stn ax + )
(A) x ci,'(- a.2) cos (ax + b)
1 .
1 . (D) cf(- ci2-) srn {ax + b)
(C) ~- a½ srn(ax + b)

8.
1 d
Particular Integral 4>(D2) cos (ax + b), where D;;;; dx and
4>(- a )
2
* O .ts ........... . (1)

1 . b
1 (B) q,(- a2) sin {ax + )
(A) 4>(- a2) cos (-ax + b)
1
1 (D) q,(a2) cos (ax+ b)
(C) x .♦ '(-a2) cos (ax+ b)

9. P-articular -Integr-al 4>(~2) cos (ax + b), wh-er-e D a :x and -4>(- a2) = 0, q,'(- -a2) ¢ 0 is ........... . (1)
1
1 (8) f(- a 2) cos (ax + b)
(A) 4>'(- a2) cos (ax + b)

l 1
(C) x 4>'(- a2) sin (ax + b) (D) x q,'(- a2) cos (ax + b)

10. Particular Integral «~2) sinh (ax + b), where D

a ddx and q,(a 2) * o is ........... . (1)

1 1
(A) 4>(a2) cosh (ax + b} (B) x 4>'(a2) sinh (ax + b)
1 1
(C) 4>(a2) sinh (ax + b) (D) 4>(- a2) sinh (ax + b)

1 d .
. -4>(a 2) '* o is ........... .
11. Particular -Integral 4>(D2) cosh (ax + b), where D • dx and (1)
1
1 (B) x 4>'(a 2) cosh (ax + b)
(A) cj,(a2) cosh (ax + b)
1
1 (D) 4>(- a2) cosh (ax + b)
(C) 4>(a2) sinh (ax + b)

x and D • -ddx is .......... ..

12. Particular lntegr al 4>(~) e V where V ls any function of
0
(1)
1 l
(B) eax -v
4>(a)
·(C) e
ax
- -1- v
4>(D + a) . (D) - - v
4>(D + a)

D s ddx is
13. Particular Integral 4>(~) xV where V is a functi on of x and (1)
(A) [ x - -LJ _l
4>{0) cj,(D) V (B) [ mJ
x - 4>(D) 4>(0) V (C) [ ·mJ
x + cj)(D) . V .(D) [ ·fml]
x- -1
q;{D) . 4>{0) V

- 1 D .:2. .
14, Particular . t
rn egra I O + - el , w here !ii dx 1s ............
1 (2)
X
(B) ee

lS. Particular Integra1--1.. .£. ·rs ........... .

_ ~ eex w here D --= -dx
0 +2e
(2)
(A) e2• .ee•
-(D) -e-x e/
16
· Particular IntegraI --1.._ • . x . h· _ .£. ·
0 - dx rs ........... .
0 + 1 sm e , w ere
{A) - e-.. sin ex (2)
(C) - e-x cos ex
(D) e-x cos ex
 ~
m {Mech. tngg. Group) CS-U)
~ATI CS - (A.6)

-GJN EER IN~~ G~M ~A~: :.::..

~EN~ :.~-- -----= ------ .::... :~--- -2!A llP£N ~·~DD(:~l'IPt.t
~-
D + -e
. Iar I nteg ral --2 1 - X
cos -eX , w heo -= dx •is ............
r-e . .£.
~llltcs
17. PartIcu
(B) e ~x sin ex (C) -..2x
• tose• (D) •"' sin•'
(A) • -• cos e' 1 (2)
2 2
. lar Integra l D 2 e- x sec x (1 + 2 tan x), (use tan x = t and
o =~
dx'
is
........... .
18. Part1cu ·+
2
(A) e-2x (1 + 2 tan x) 2
(a) e- x (tan X + tan 2 X) (2)
2 2
(C) e x {tan x + 2 tan x) (D) -2x (
. e tan x + sec x)
1 ( 1 ) d .
19. Particu lar Integral O + 1 1 + ex where D - dx Is ............

(B) log (1 + ex) (C) ex log (1 + ex) (2)

(A) ex log (1 - ex)

20. Particu lar IntegraI o f d ·tt . I equa t·ion dx2 - 7 dx + 6y = e2x Is

I erentIa
• ~ ~
xe2x e2x e2x
(A)
3 (B) -4 (C) 4 (D)
.e 2x

24
21. Particular -Integra l of diff-er-ential -equation (0 - 50 + 6)
2 y = 3e5x is
e sx e5x (2)
esx
(A) 2 (B) -
6 (C) (D)
e2x

14 2
22. Particular Integra l of differe ntial equati on (D
2 - 9) y = e3x + 1 is .......... ..
(2,
1 e3x 3 e 3x 1
3x (C) x-6 -- -9 l-
(A) --e3x_
2
_
9
(B) x-6 + -8 (D) xe3x +
8
2 x is .......... ..
23. Particu lar Integra l differe ntial equati on (D + 4D + 3) y = e-3
(A) xe- 3x 1 X e-3x
(B) - - e-3x (D)
__
2 2
24. Particular Integ.ral of differe ntial equati on (0 - 2) 3 y = e2 x + 3x is ............

( 0 1 (B)
0
- e2x + 3 3 3x
1
- -3 3x
A) - e2x - -3-2) 3! (e-2)
3! +(log
X 2x 1 x3 1
(C) 3! -e + (log 3 - 2) 3 3x (D) 3! -e2x + (log 3 - 2)3
is ............
25. Particular Integra l of differe ntial equati on (D - D) y = 12ex
5

12 (C) 12xex (D) 3xex

(A) 3ex (B) - xex
5
26
• Particular Integra l of differe ntial equati on (D2 + 1) (D - 1) y =
ex is

(A) xex 1 1 (D) x2 e'

(B) -x2 ex (C)
2xex
2
27
· Particular Integra l of differe ntial equati on (D2 - 40 + 4) y = sin 2x is .......... ..
cos 2x
(A) _ co~ 2x (B) ~ (C) sin/' =--
(D) X 8
8
I f d. 3 ·
28. Particular Inte
+ D) y = cos x Is ............
gra o · 1fferen tial equati on (D
X
(A) 1 -2cos x
! . X
(() - - cos X
(D)
- 2 Stn X
I •
(B)
4 cos X . .
2
29. Particu larlnte
(D 2 + 1) y = sin x Is .......... ..
gra of differe ntial equati on 1 'X
X
(D) - 2
-cos
(A) - 2 cos x x (C) _! sin x
{B) -
4 cos x 2
 ApPENDIX : MULTIPL"f CHOICE QUESTIONS
(A.7)
£MATICS m (Mech. "tngg. Group) (5-II)
GINE£R1NG MATH - . (2)
EN . 3 + 90) y = sin 3x Is ............
30 Particular Integral of differential equation (0 . 3 1 . 3
• X .
(0) - 18 sin x
(() - X sin X
x (B) - - sin 3x
(A) - - cos 3x 18
18 . (2)
. x + cos 4x Is .......... ..
. (04 + 1002 + 9) y = sin 2
31 Particular integral of differential equation 1 .
• 4
(-B) - sin 2x + cos x
1 _L . 15
(A) -T3
sin 2x - 105 cos 4x
1 1
(0) - 15 sin 2x + 87 cos 4x
(C) - 115 sin 2x + 1~5 cos 4x

. I ~ Qi: + Sy - .. (2)
ation 2 10 sin x IS
32 . Particular Integral of di fferen t ,a equ
..... .. .. .. .
dx2 - dx -

8 . .(B) sin x -2 cos x (C) 4 sin x + 2 cos x (D) 2 sin x+ cos x

(A)
3 sin x
33. Particular Integral of differenti al equation (04 - m 4 ) y = cos mx i-s .. ......... . (2)
-x 25... . -x
(D)
(A) 4m cos mx (B)
m sin mx
(C) - x sin mx ~sin mx
-3 3
m
... ~ ~
34, Particular Integral of d1fferent1al equation dx3 - 4 dx = 2 cosh 2x .1s ........... . (2)
1 X X X .
(A)
4 cosh 2x (B)
8 cosh 2x (C)
4 cosh 2x (D)
4 smh 2x
35, Particula rlntegral of differenti al equation {02 + 6D - 9) y = sinh 3x is .......... ..
(2)
1 1 1 . 1
(A) cosh 3x (B) cosh 3x
18 2 (C)
18
. s1nh 3x (D) -
18 cosh 3x
. Iar I ntegra I o f d·tt
36. PartIcu I·. erentIa . ~
. ·1equation dx3 + 8y = x4 + 2x + 1 Is
. ........... .
(2)
1 1
(A) - (x4 + Sx + l) (B) - (x3 - 3x2 + 1) 4 1
8 8 (C) X - X + 1 (D) - (x4 -x + 1)
8
37. Particular Integral of different ial equation (04 + 0 2 + 1) y = 53x 2 + 17 is .......... ..
(2)
(A) 53x2 + 17 (B) 53x2 - 89 (C) 53x2 + 113 (D) 3x2 -17
38. Particular integral of different ial equation (D 2 - D + 1) y = 3x2 - 1 is ............
(2)
(A) 3x2 + 6x + 5 (B) x 2 - 6x + 1 (C) 3x 2 + 6x - 1 (D) 2
x + 18x- l l
39. Particular Integral of different ial equation (D 2 - 1) y = x 3 is ........... .
(2)
(A) - x3 + 6x (B) x2 + 6 (C) x 3 + 6x 3
(D) - x - 6x
40. Particula rlntegral of different ial equation (D 3 + 3D 2 - 4) y = x2 is
(2)
41(x2 + 23) 41(x2 23) (x + ½)
(A) -

41• Particular Integral

(B) x+ (C)
2
(D) -¼(x
2
-½)
of different ial ·equatio n (D4 ·+ 25) y = x4 + x2 + 1 is ........... .
(2)
(x + x (x + x + ~~)
4 2
(A) - \) (B}
4 2
2

(C) 2\ (x4 + x2 + 24x + 1) (D} 2\ G4 + x2 + 2\;)

42
• Particular .Integral of different ial equation (D 2 - 4D + 4) y = e2" x4 is .... ....... .
x6 (2)
(A) 120 e2• (£)
x6
-e2• x6 XS
60 (C} - e 2" (D) - e2•
30 20
 Af'PENDlX : MU\.TIPlE CHOlC£
Q\J lST I:~
!~-~-~A~n~cs~-~m~CMec~~ch~-~tngg~~-G~rou~p
EN
~'°"G~~•""".. ~~_!{S~-~m~--~(~A.:.: !:ll!..-------
1of diff ere ntia l -eq ua tiod2
n~
d - . (2)
partiCUlaf Int-egra + 2 .QY + y = -e x cos x 1s .......... ..
43. dx- dx
(B) - e-• sin x (C) - e-• cos x
(A) ex cos X
_ particular inte gra l of diff ere (2)
ntia l equ atio n (0 2 + 6D + 3 3
44 , 9) y = e- x x- is .......... ..
e-3x
(B) e- 3x x
(C) 12X
45. Particular Inte gra l of diff (2)
ere ntia l equ atio n (02 + 2D
+ 1) y = e-x (1 + x2) is ........... .
(A) d~ -~i) (B) .~ ( . . ~ (Cl .-• ~ fi) +
(2}
46. Par ticu lar Inte gra l of diff
ere ntia l equ atto n (D - 1) 3
y = ex ~ is .......... .. 3
4
8 (D) 8 e• x-5/2
15 e• x5
(A) - 12 712
105 e• x
(8) -

4'1. Par ticu tar inte gra t of diff i d (2)

ere ntia l eq ua tio n~ - 2 .QY
dx dx + y-= xe• sin x is ...... .... ..
(A) - e• (x sin x + 2 cos x)
(8) e• (x sin x - 2 cos x)
(C) (x sin x + 2 co-s x)
(D) - e• (x cos x + 2 sin x)
· n o f d I'ffere ·
48 • So IutIo · n dx2 + .9Y
ntta l equ atio dx + y = e2x 1ss!:Y
• ......
.... ..
(2)

(A) e• ( C1 cos "1 x c sin"1 x)-½ e

+ 2
2
•
(B) e
½ X (
c1 cos
-Ji x + sin, ·:ii ) 1
2 .c:2 x + 5e 2
2X

~
.l x
(C) e
2
c
( 1 cos ½ x + c2 sin ½ x) + ½e• (D) e
-½ X (
C1 cos
,
2 X + C2 Sin 2
-ii X)\ + 17 e 2x

49.. Sol utio n of diff ere ntia l

equ atio n (D 2 + 1) y = x is
...... .... .. (2)
(A) C1 cos X + C2 sin X - X (8) C1 cos X + C2 sin X + X
(0 C1 cos +

-
X C2 sin X + 2x (D) C1 cos X + C2 sin X - 2x

1. (A) 2. (0 3. (D) 4. (D) 5. (C) 6. (8) 7. (B)

9. (0) 10. (C) 11. (A) 12( C)
8. (A)
ll. (0) 14, (A)
17. (D) 18. (B) 19. (D)
15. (B) 16 . (C)
20. (B) 21. (A)
25. {D) 26. (C)
22. (C) 23. (D)
27. {8) 28. (0) 24. (A)
29. (A) 30;(8)
33.(D) 34. (C) 35. (A) 31. (C)
36. (D) 37. (B) 32.(D)
41. (D) 42. (C) 38. (C)
43. (0 44. (A) 39. (D) 40 . (A)
49. {8)
45. (C} 46. (B)
47. (A)
Typ e m : Me tho d of Variatl 48. (D)
on of Parameters :
1. Co mp lim ent ary fun ctio n 2
. . . of d1ff
ao il
erent1al equ at'
. ion dv
dx2 + a1 .:.z..
par am ete rs, par ticu lar inte + a2 = . ·
gra l is u(x, y) Yi + v (x
f(x) . Marl<s
~ 'y) Y2 where u dx Y f(x) IS C1Y 1 + c2y 2
is obtained f The
< f Y1Y2 +. Y2Y1 ~ · n by me tho d of variation of
mm .•..•.......
(C) f y2 f(x) (B) J Y1 f(x) (1)
Y1Y2 -Y2Y1 Y1Y2 - Y2Y1 dx
(D) f - Y2 f(x)
Y1Y2 - Y2Y1 dx
~ MA11ffMAl1CS _ m(Mech. Engg. Group) (S-U) (A.9) ~PPENDIX : MULTIPU CHOICE Q\JISl10NS

~ ·
iementary function of di
'ffer . I
,ent,a equation
• . 80
£ty _gy .
dx2 + 81 dx + fliY = f(x) is C1Y1 + Cit,. Then by method of variation of
C()(TlP
Z. ters particula r integral is u(x, y) Y1 + v(x, y) Y2 where v is obtained from ........... . (1)
pararne
-1' f(x)
•
dx (B) f - Yi f(x) dx
~ f ~-~1
~-y~
-y, f(xL dx (D) f Y1Y2f(x)- Y2Y1dx
(Cl f y y2- Y2Y1
1

J. In solving different ial equation ~ + Y = cosec x by method of variation of parameters, complimentary function = c cos x + 1

(2)
ci sin x. Particular Integral = u cos x + v sin x then u is equal to ............
(B) (C) - X (D) log sin X
(A) - log sin )( X

~
.. .
In solving d1fferent1al equation dx2 + 4y =sec 2x by method of variation of parameters, complimentary function = c1 cos 2x
+ c, sin 2x, Particular Integral = u cos 2x + v sin 2x then u is equal to ............ (2)
l l
(Al -
1
2 x (B) 4log (cos 2x) (C) -
4tog (cos 2x) (0) (½) x
s. In solving differential equation ~ - y = (1 + e•xr 2
by method of variation of parameters, complimentary function = c1e• +
(2)
cie-•, Particular Integral = ue• + ve·• then u is equal to .......... ..
(C) log (1 + e•)

6. In solving diffe.rential equati on~ + 3

2
t + 2y = sine• by method of variation -0f .parameters, .complimentary functi-On
= c1e·•
(2)
+ cie-2x, Particular lntegraJ = ue·• + ve· • then u is equal to ·····-··-·
(B) - cos (e•) (C) cos (e•) (D) ex sin (e•) + cos (e•)
(A) - e' cos (e') + sin (ej
:gy
7. In solving differential equation dx2 - 6 dx + 9y =
~
7 3x
by method of variation of parameters, complimentary function = c 1xe
3
•

3 3 (2)
3
+ c2e •, Particular Integral = uxe • + e • then u is equal to ............
l
- log x
(A) - 1;X (B) lX (C)
X
(D)

l In solving differential equation ~ + y = tan x by method of variation of .parameters,

complimentary function = c 1 cos x + c 2
(2)
sin x. Particular Integral = u cos x + v sin .x then v .is equal to ............
(A) - cos x (B) [log (sec x + tan x)] - sin x

(C) - [log (sec x + tan x)] + sin x (0) cos x

of parameters, compiimentary function -= Ci cos

9. In solving differential equat ion~+ 9y.:: 1 + ~in 3x by method of variation
3x + c2 sin 3x, Particular Integral = u cos 3x + v sin 3x then vis equal to ........... . (2)

(A) i (-½sec 3x+½t an3x-x ) (B) -¼log( l+sin3 x)

1 1
(C)
9109 (1 + sin 3x)
(D) 3log cos x
_g,:y 2 b . . of parameters compl'imentary f ct·
hOd Of variation x
10• 1n so·1ving
•
differential equation dx. .2 - y = -1 + ex Y met ' un I0n = c1e + c2e...\

Particular Integral = uex + ve·• then v is equal to ...... ...... (2)

(A) e-• - log (1 + e·•) (8) - log (1 + e•)

(C) log (l + e•) (D) - e·• + log (1 + e·•)

_....-.rrftft..lG MAllf!MATICS - lD (Mech. 'Engg. Group} (S-U)
i::,
.." ....i::~~•" (A.10) AP--
CH<>Ict
~
r~rn,&A: M\ILTIPL£

11.
In solving differential -equation
QY
dx2 + 3 dx
X
+ 2y = -ee by method of variation of parameters,
~
complimentary f .
c e-2x + c e-x, Particular Integral -= ue-ix + ve-x then v is equal to ........... . unction
1 2
X

(A} - ee
X
e-2x ee X
(8) (D) e'
d2
. In solving differential equatio n~+ 4y = 4 sec 2 2x by method of variation of paramete
12 rs, complimentary function_
- C: COS
2x + c sfn 2x, Particular lnt-egral = u cos 2x + v sin 2x then v is -equal to .......... ..
2
(2)
(A) log (sec 2x + tan 2x) (B) - sec 2x (C) sec 2x + tan 2x (D) log (tan 2x)

1. (D) 2. (A) 3. (C) 4. (B) 5. (D) 6. (B) 7. (C) 8. (A}

9. (C) 10. (B) 11. (D) 12. (A)
Type lV .: Cauchy'.s .and .Legendre'.s linear Differential .Equations.:
1. The general form of Cauchy's linear differential equation is ........... .
(1)
(A) a0
~
dxn + a1 ~
dxn-1 + a2 ~
dxn-2 + ... + any = f(x), where a0, a1, a2, ... , an are constants.

dx :91 dz .
(B) i,- = Q = R, where P, Q, R are functions of x, y, z.
dn d"-ly dn-2
dlCn + a1xn-1 dxn-1 + a2 xn-2 -
(C} a0 xn .Q.:t - + ... + any = f( x), where ao, a1, a2
dxn-2 ... an are constants
d" dn- 1 dn-2
dxn -+ a1 (ax -+ b)n-1 ~
(D) ao (ax-+ b)n.Q.Y dx~1 -+ a2 (ax-+ ·b)n-2
· ~
dxn-2 -+ ... ·+ ariy = f(x), wh· ere a0, ai, a2 ... , an are constant.
-d" d"-1 dn-2
2. Cauchy's linear differential equation a0x" ~ + a.ix"-1~ + a-2xn-2~ + ... + any = f(x) can be reduced to linear differential

equation with constant coefficients by using substitution ............

2
(ll
1 1
(A) X= e (B} y = ez (C) x = log z (D) X =e
3. The general form of Legendre's Hnear differential equation is .......... .. (1)

s!:Y dn-ly _ an-2y . _

(A) ao dx" + a1 dxn-1 + a2 dxn-2 + ... + any = f(x), where a 0, a 1, a 2 ... , an are constant.

dx :gy dz .
(B) p = Q = R , where P, Q, R are functions of x, y, z.
d" dn-1 dn-2
(C) aoxn2:Y
dx" + a1X n..:1 ~
dxn-1 + a2Xn-2 dxn-2
· + ... + any = f (X) , Where ao, a1, a2 , · ., 3n are constant
d"
(D) ao (ax + b}" ~ + (ax + b)n-.i ~ d"-1 dn-2
31 + a (ax + b)"-.2 ~ + ... + any = f(x), where a0, a1, a2, ... , an are con 5tant ·
2
..in-.i..,
I • • •
4• Legendre s linear d1fferent1al equation a (ax + b)" d n +
~ n-1 ,!L_..Y.
dn-2 .
b)n-2 g::_:j'.2 + ... + any = f(x) can oe
0 a 1 (ax + b} X dxn-1 + a2 (ax + dxn-
(U
reduced to linear -differential equation with constant coefficients by .using subS t itution ............
2
(A) x = ez (B) ax + b = ez (C) ax + b = log z (D) ax + b = l
S T . t coefftcit>f1ts.
d d2v dv ff ·I
• 0 re uce the differential equation
x2 .:::...1,.2_ 4x .::L + 6y = x4 to linear di erentia equation with constan (l l
dx dx
substitutions is .......... ..
(A) X = z2 + 1 (D) i = log z
(B) X = e1 (C) x = log z . h constant
6. To reduce the differential equation (x + 2)2 !¥- (x + 2) ~+y = 4x + 7 to linear different
. ial equation wit
(1)

coefficients, substitution is ........... .

(D) x -+- 2 = log z
(A) x + 2 = e-z
(8) X = Z + 1 (C) X + 2 = e'
ENGINEERING MATHEMATICS - m (Mech. Engg. Group) {S-R)
(A.ll.)
To reduce the differe f . ,.,.. APPENDIX , MUI.,... cHOICl qutSl10NS
7. n ,a 1 equation (3x + 2)2 .:.!..X. .Ql
. . dx2 + 3 (3x + 2) - 36 - i
constant coeff1c1ents, substitution is .. dx Y - x + 3x + 1 to linear diff-erential equation with

(A) 3x + 2 = e1 (1)
. (B) 3x + 2 = z (C) x = e• (D) 3x + 2· = log z
(1)
a. d
. D = dz the differential equat'ion x2 dx2 + x .Qi'
On putting X -- e1 an d using dx.. + y = x is transformed into ~
1
(D) (D 2 + D + 1) y = e
2
(A) (D - 1) y = e1 (B) (D 2 + 1) y =e' (C) (D2 + 1) y =x
d.
x2 ~-
x = e1 and using D • dz ,s transformed
9. .The differential equation dx2 x .Qi'
dx.. + 4y = cos (log x) + x sin (log x) on putting
(1)
into ..... ....... '

(02 - 02 + 4) y = sin z + e' cos z

C) (0
(A) (B} (D 2 - 2D + 4) y = cos (log .x) + x sin (log x)
1
(
2 + 0 + 4) Y = cos z + e-z sin· z (D) .(D2 - .2D + 4) y = cos z+ e sin
. z

~
(1)
10. On putting x = e'Z the transformed differential equation
. of x2 d 2- d .. + Sy = xi sin
3x .Qi' . (log x) using
. D =..Q....
d 1s ........... .
2 X X Z
2
.
(A) (0 - 40 + .5) y = e2z sin z (B} (D - 4D + 5) y = x2sin (log x)
2
2 1 (D} (D - 3D + 5) y = e• sin z
(C) (D - 4D - 4) y = e sin z 2
(1)
x2 ~
d2 d 3 .
11. The differential equation 1 + -x2· on putting
dx _ Y -_ _x
dx2 + x 2.l . x = ez and using
• D = dz
d .1s transformed
. .into ........... .
e3z
x3 (B} (D 2 - 2D - 1) y = l + e2z
(A) (D 2 - 1) y = -1 + X
2
3
e'Z
e3z (D) (D 2 - l)y = - i 1
(C) (D 2 - 1) y = -1 +e2z 1 + ·e
(1)
12. Th e d'ff
r erentra · x1 Jl:x
· I equation dx2 - EY .
Sx dx + Sy = xl log x, on putting
· x = e• and using
. D = dz
d .1s transformed .into ........... .
2
(A) (0 2 - SD + 5) y = z ez2 (B} (D - z
5D - 5) y = e'z
2
(Q (D 2 - 6D + 5) y = x' log x (D) (D 6D + 5) p z e"

~ ~
-

13. The differential equation (2x + 1)' - 2 (2x + 1) - 12y =6~ on putting 2x + 1 =e' and putting D • 1z ts transformed(1)
into ............ (B) (D 2 + 2D + 3) y = 3 (ez -1)
(A) (D2 - 20 - 3) y = ¾(ez - 1)
(D) (D 2 - 2D - 3) y = 6x .
i l z 1)

~
(C) (0 + 20 - 12) y ·= 4 (e -
~ ·2Y _l 3 2 2 • 3 . z
14. The differential equation (3x + 2) 2 dx' + 3(3X • 2) dx - 36Y - 3 [( x • ) - l] · On putting x + 2 = e' and using D dd is
(1)
transformed into ........... . (B) (D2 + 4) Y = 91 (e2z - i)
1 2
(A) (D 2 + 30 - 36) y =
27 (e z- 1)
(D) (0 2 - 9) y = (e
2
z- 1)
2
(C) (D 2 -4) y = } (e z-l)
~
7
lS. The d'1ffe t· I t· (1 + x)2 d2v + 3(1 + x) .9Y
dx - 36y = 4 cos [log (1 + x)] on putting 1 + x = ez and . d
ren ,a equa 10n dx using D " dz is
(1)
transformed into ........... . (B) (D2 + 20 - 36) y-= 4 cos z
(A) (D 2 + 20 - 36) y = 4 cos [log (1 + x)] (0) (02 - 20 - 36) y = 4 cos (log Z)
(C) (D 2 + 30 - 36) y = 4 cos z
 APPENDIX : MULTIPLE OfO IQ
QUlsnONs
(A.1 2)
di. Engg. Gro up) {5-D)
ENGJNEERING MATHEMATICS - m {Me
~ . ..£.. . t
4x + 1 = ez and using D • dz 1s ransfol'flled
1)
2~
dx2 + 2 (4x + 1) dx + 2y = 2x + 1 on putting
diff-erential -equation (4x +
1~. The (1)
int-0 ............
(16 0 + ·8 0 + 2) y = (-ez + 1)
2
1 z (B)
(A) (D
2
+ D + 2) Y = 2 (-e + 1)
(0 + 20 + 2) y = (ez - 1)
2
1 - (D)
8D + 2) y = (ez + 1)
(C) (16 D -
2
2
2 and using D • .£!. is
+ 2)] on putting x + 2 = ez
d
(x + 2)2
d
~ + 3 (x + 2) ~ + y = 4 sin [log (x dz
17. The differential equation (1)
transformed into .......... ..
(B) {0 + 1) y =4 sin z
2

(A) (D + 30 + 1) y =4
2 sin (log z)
(D) (0 + 2D + 1) y = 4 sin
2 z
2
+ 2D + 1) y = 4 s·in Tlog (x + 2)1
(C) (0
by .......... .. (2)
2-~ + x -~ + y = x2 + x-2, complimentary function is given
atio n x
18. For the differential equ
(8) C1 log X + C2
(A) C1X + C2 (log X)
(D) ·c1 cos (log x) + C2 sin
(C) C1 cos X + C2 sin X
(2)
. l equation ~ l_gdxy =A + Blog x, comptrmentaryfunction is given by .......... ..
1'9. 'For the differentia dx2 +;
:fl + C2
(D)
X

(2)
ction is given by............
20. For the differe ntial equatio n x'2 - ~
4x -~ + 6y = /', complimentary fun
5
C1X-2
+ C2X-
3 (0) C1X + C2X
(C)
3 (B) c1x2 + C2X
(A) C1x2 + C2X (2)
given by ....•.......
x), complimentary function .is
2L For the differential equatio.n
.x2 ~ ~
-x + 4y = cos (log x) + x sin (log

(B) X [ C1 cos (log X) + C2 sin V2

(log x)] 12
f
(A) C1 cos ,{3 (log X) + C2
sin ,{3 (log X)]
(0) X ( C1 cos {3 (log x) +
C2 sin {?, (log x))
(Q X [ C1 cos (log x) + C2 sin (log x)]
(2)
d 2u -du ction is given by ...... .... ..
- u = - kr , complimentary fun
3

22. For the differe ntia l equ ation r2 +dr

r dr

(A) (c1 log r + C2) r

~
r)] (D) Cir
.:i
+ r
(C) fc1 ·cos (log r) + c2 sin (log (2)

2~ . n by .......... ..
· Iar ·rntegra I ·,s give
.ff t·
,a I equa t·
ron x dx2 + x .9Y
dx + y = x, partrcu
23• For th e d , eren
X (D) 2x
(B) ! (C) -
(A) X 2 3 (2)

d2 d l is given by .......... ..
n x2 , 4x ~ + -6y = xs, particular integra
24. For the differential equatio
_

XS x4 (D) - 44
t..
(A) 6
XS
(.B) 56 {C) 6 (2)

25. Solution of differe~tial equ

ation x ~ + -~ = xis ...... .... ..
x2
c2) -
x2
{B) (c1x2 + C2) + 4
(A) (C1X + 4 x2
(D) (c 1 log x + c2) + 4
 TIONS
IPLE CHOICE QUES
THEMATICS_ m{M
ech E APPENDIX: MULT
ENGINEERING MA · ngg. Group) (S-11). (A .13 }
(2)
~ + x .Qi: _ .1 .
en tia l eq ua tio n X: dx2 2 dx - X: ts .. .... .. .. ..
26 . Solution of dif fer
X:
(A) (C1X + C2) -
4
x2
+
(D) (C1 log X + C2)
4
.......... <2>
tary fun ction is given by ..
~ + (x + 1) ~
+ 1)], complimen
+ y = 2 sin [log (x
27 th dif fer en tia l eq ua tio n (x + 1)2
• For e l)]
+ ei s.in [lag. (x +
(B) c.1 c.as. [lag (x + l)J
(x + 1) + C2 (x +
1r1
(A ) C.1 (log x)
1) (D) c cos (log x) + c2 sin
1 (2 )
(C) fc1 log (x + 1) + c 2 ] (x + . • b .... ...... ..
2
function- ~s gwen- ·
. Y .
3)2 ~ ~
dx - 12 y =- 6x, co mp hm en tar y
.
tia l eq ""' t· (2 dX: - 2 (2x + 3-)
28. Fo r th e dif fer en ....,.. ion x +
3
(2x + 3r + C2 (2x + 3)
1 (8) C1
3
(2 x + 3) + C2 (2X + 3r 1
(A ) C1 - 3r
(D} C1 (2X - 3)2 + C2 (2X
3 + 3)2 (2 )
(C) (2 x + 3) + c2 (2x en by .... ........
tary fun cti on is giv
c1
+ 2)2, co mp lim en
nt ial eq ua tio n (3x
+ 2)
2
~ + 3 (3x + 2) ~ - 36 y = (3x
29 . For th e dif fe re
2
+ c2] (3x + 2r
3
(B} {c1 log (3x + 2}
(3 x + 2)3 + C2 (3x + 2f c (3X - 2r
2
(A) C1
2
(D) c 1 (3x - 2)2 + 2
2
(3x + 2f (2)
(3x + 2) + is given by .. ........ ..
C2
(C) C1
limentary fun cti on
ua tio n (x + 2)2 ~ -
(x + 2) ~ + y = (3x + 6}, co mp
ial eq
30. Fo r th e dif fe re nt
(B) c1 fog (x + 2) + c2
1
(X + 2) + C2 (x + 2f (x + 2)
(A) C1
[c1log (x + 2) + c2J

-
1
(D)
(C) C1 (X - 2) + Cz (X - 2r

7. (A) 8. (8 )
5. (B) 6. (C)
3. (D) 4. (B)
2. (A) 15 . (B} 16. (C}
1. (C) 13. (A) 14. (C)
11. (C) 12. (D)
10 . (A ) 23. (B) 24. (A)
9. (D ) 21. (D) 22 . (B)
19 . (C) 20. (A)
17. (D) 18. (D) 30. (D)
28 . (A) 29. (C)
26. (C) 27. (B)
25. (D) •••
l E ua ti on s
m m et ric Si m ul ta neous, D iff er en tia
l E uations, S
Linear Differentia Marks
Ch. 2 Simultaneous
nt ia l Equations :
ous Un ea r Di ffe re
Type I : Simultane
tions ........... .
an eo us lin ea r dif fe re nt ial equa
l For the sim ult
·1s o.,h ,ta·in f rom (2)
·ing D =.£ dt
d 2 - e t solut
2 ion of x us
-dx 2 ~ 3x + Y -
dt + x - 3Y = t, dx - 2
(8) (0 - 4D - 5)
x = 1 + 2t - 3e2t
= 1 + 2t + 3e2t (0 + 4D - 5) y =
2 3t + 4e2t
(A) (D + 4D - 5) x
2

2
(C) (0 + 4D - 5) x
= 3t + 3e2t
(D)
(
us e D ,. :t)
~ 2 - 3Y=t, ~
of x re su lts in
dt -
3x + 2 Y = eit eli m ina tio n
2 For the system of linear dif fe re nt ial eq ua tio ns dt + x
· (2)
2
(0 - 4D - 5) y =
t - 4e2t
····· ······· (B)
+ 3e2t 3t + 4e21
= 1 + 2t (0 + 4D - 5) y =
2 2
(A) (D + 40 - 5) x (D)
21
40 + 5) y = 3t - 2e
2
(C) (D -
 MATHEMATICS - m(Mech. Engg. Group) (S-0) (A.14) APPENDIX . MUL
ENGINEERJNG
· TIPU CHOI(! QIJ
. du . dv . . d ~
_ For the simultaneous Linear DE dx + v = sin x, dx + u = cos x solution of u using D ■ dx is obtain from .......... ..
3
2
{2)
u = 2 cos x (B) (D - 1) u = 0
(A) (D2 + 1)
2
(C) (D 2 - 1) u = sin X - cos X (D) (D - 1) V = - 2 sin X

. x, ~ d ) .......... ..
. . u resu Its in ( use D ■ dx
dx + u = cos x eI'Immatmg
For the simultaneous L'mear
DE du
dx + v = sin
4.

(A) (D 2 + 1) V =0 (B) (D 2 - 1) u = 0
2
(C} (D 2 - 1) V = - 2 sin X (D) (D + l) v = sin x + cos x

s. For the simultaneous Linear DE ~: - 3x - 6y = t

2
, ~ + ~~ - 1
3y = e. solution of x using D a :t is obtain from .......... ..

(A) (D2 + 9) x = 6e
1
-
2
3t + 2t (B) (ct + 9) y = - 2e1 - 2t
1 2 2 1 2
(C) (D 2 - 9) X = 6e - 3t (D) (D + 12D +- 9) x = 6e + 3t + 2t
dx .2Y
6. For the simultaneous Linear DE L dt + Rx + R(x - y) = E, L dt + Ry - R(x - y) =0 where L, R and E are constants, solution of x
d
using D;. dt is obtain from .......... .. (2)

2
(A)
2
(L2D2 +- 4RLD + SR ) x = 2RE + 2R (B) (L2 D2 + 4RLD + 3R ) y = RE
2 2
2 2 2
(C) (L D + 4RLD + 3R ) x = 2RE (D) (L2D + 2RLD + SR ) x = 2RE
. . dx ,2Y of y
7. For the simultaneous Liner DE L dt + Rx + R(x - y) = E, L dt + Ry - R(x - y) = 0 where L, R and E are constants, solution
d
using D = dt is obtain from ............ (2)

2
(A)
2
(L2D2 + 4RLD + SR ) y = RE + 2R (B) (L 2D2 + 4RLD + 3R ) y = RE
2
2 2
(C) (L D + 4RLD + 3R
2
) x = 2RE (D) (L2D2 + 2RLD + SR ) y = 2RE

· Itaneous L'mear DE dx
8• For t he sImu
1
dt + x = e· so IutIon
dt + y = e , .2Y
1 · of x using dt ·Is obt am
· D = .£. · f rom .......... .. (2)

2
(A) (D - 1) X = 2e
1
(B) (D 2 - 1) y = - e1 - e· 1
1
(C) (D 2 + 1) x = e· + et (D) (D 2 - 1) x = et - e·1

dt ·Is o btam
· D. =.£. (2)
dt + x = e· , so Iut·I0n of. y using
dt + y = e, .2Y
· f rom .......... ..
1 1
9• · Itaneous L.mear DE dx
From t he s1mu

(A) (D2 - 1) y = 2et (B) (D4 - 1) y = -e1 - e· 1

1

* (D 2
2
(C) (D + 1) y = e· 1 + e1 (D) - 1) x = et - e·

10. For the simultaneous Linear DE ~~ + Sx - 2y = t, + 2x + y = 0, solution of x using D = :t is obtain from .......... ..
(21

2
(A) (D + 6D + 9) x = l + t (B) (D2 - 6D + 9) x = 2t
2
(C) (D + 6D + 1) x = t (D) (D2 + 6D + 9) y = 2t
(2'
11. For the simultaneous Linear DE~; + Sx - 2y = t, ~ + 2x + y = 0, solution of y using Da :t is obtain from .......... ..

(A) (D
2
- 6D - 9) y = 2t (B) (D 2 + 6D + 9) x = 1 + t

(q 2
(D + 6D + 1) y =t (D) (D 2 + 6D + 9) y = - 2t

7
, (~D)L_.!-....:3~-~(B~)---l-~~_L ~~!__J_, ..:6:::...·~(C~). - L --· _(B) __.___
. (A~)~t-..f:.2:.J
r-~l~
9. (B) 10. (A) 11. (D)
-tN G IN llR IN G M A T
H E ~n cs - m
1. ty
Thpe ege
Une
·· 5raym
(Mech. En gg

~
l fomrm . l G
etrt
ofcasymSim.ult..an eo u . ,o up )( S· U J
s Differential_E
metnc nuat·

~
g'.'.Y si I (A.15)
(A) ao dxn +
d" ·' . u,ns . APPtNDIX: M
a1~ ,-d, + a d" -'y
m u taneous DE . ULTIPLE cHot
ct n,ut s..,oNS
dx d dx . .
(B) - = !!l'. - 2 dxn-2 + . .. + ,/J " ............ "
3. a - f(x), where a,
a a
p
d"Qv - R , wheo-re p' Q , R are fu nc
' '' ' .... a, are co
nstant M•'"'
> f \11
~ of X, y, z
10 n
(C) aoXn
(D ) Xn-,1 st
. dxn + a1a~
d..=
Xn-Y.
a (ax + b)° "- '- 1 dn
+ a2xn-2 SL -2:J'.
dx....
n -2 + ... + a,y= f(x
d "- ' ), where a a
2 Solut0ion f
. dxn +. a1- . (ax + b r l SL....: • " a, .. " a, ar
e constant
dxn-J'_1d.+ a2 (ax + b
r2 dn -2y
dx"·' + ... +.a,y •
•
symmetrn:: sim
f(x), where a,, a,,
ultaneous 0£ a, .. .. a, are consta
(A) X + y0 0,
y+z 0 = = - 9Y - dz .IS
.21. _ nt
(C) X + l - l .. ..........
y = Co y - z= C2
. X/ -• zc,= C.1,
(B) X' \11
~
, yz •y c,
- z= C_2
3. Solution of
sy mmetric simulta (1)
neous DE chX< = !!1y. -- dzZ 1s .......... ..
.
(A) )( : C1Y, Y (B)
= C2Z X - Y =- C1Z,
y - Z =- CzX
. ) X + Y : C1,
(C (D)
Y+
Z = C2 . X+ Y = C1, Y - Z =
• symmetrical si C2
4. Consideri
ng th e first two
ratio of th
multaneous DE 7 = ,! =H i ,one of the re
. . lation in the solu
dx l!'i tion is OE
(1)
IS ... .. ... ... . dz
1 1
(A) - - - (B) (C) x2 - I= (D) x3 - l =c
y =c
X
X- y=c C
5. Considering y -X'/ x(Z _ ly ), one of
th e first two ratio of the sym the relation
metrical simultane 2 in the solution
of DE is ............ ous DE ·dx =2i- = dz

(8) x 3 (0 ) x2 - / = c
1;. (A) x2 + y2 = c
Con<ide<ing th e firs
+ y3 = C
t two ratio of th dx 9J_ dz (2)
e symmetrical sim
ultaneous DE 7z =,!z =fX ,one of the rela ·
tio n in th e solution (2)
(A) x'
is ... ... ... - / =c of OE
. .. (B) x - y= c
7. Considering (Cl x' - l =c
th e first and th
ird ratio of th -l + y' =c

(A) )f -
e symmetrical si
multaneous DE
= 7z i 7, =
(D)

one of the relatio

n in th • solutio(2)
of
DE is ........... .z' =C n
8. Considerin9
(B) x' - y' =C .
th e second .(C) x' - z' =C

~
and third ratio of th dx (D) j:j
X,_ z=dzC
e symmetrical sin, -
ultaneous DE ,! - -I - z' = 2-,,; = ' one of th e re (2)
the solution of lation in
DE is ........... •
(C) y = a lD) X- z=C
(A) .12. 1. (B) f - z2 = c
2=
z t ofc .
-
y a se
9. Using ~-.21--~'
multiplier as 1. (2)
1, 1 th • solutio
n of DE y - z -

~
(Al ,! + -I + z - x - x - y is ...
z' • c ........ . (0 ) - X + Y
- Z=C
(B) , _ y - z
lO. Using a set . c ~ -_E d x-
(C) L
of multiplier as +-y + zdz= c
x, y, z th • solull00 . (2)
of DE 3z - 4y -
4x - 2z - 2y - 3x
is ...... ... ...
1 l 1-c (C) X + y + Z = (0) x2 + / + 2
(B) ; + y + C z =c
(A) x3 + y3 + z-
z3 = c
 {A.16) APPENDIX : MULTIPLE CHOICE QUEST10Ns
ENGINEERING MATHEMATICS - m {Mech. Engg. Groupl (5-D}
dz. . (2}

ltip lier as x , y , z
3 • ,3
t e so
I .
ut1
3
on ohf dx.
DE x(iy 4 _ z4) =y{~ dy
_ 2x4)- =z(x4 _ y4) 1s ........... .
ll. Using a set of mu
(D). xyz = c
(B)
4
x +y +z =c
4 4 (C) x+y+z=c
(A) x3 + y3 + z3 = c (2)
. .. . ~ -=£!l dz -.
-x = ix _ y 1s ••·· ... •··· ·
12. Usmg a set of mult1pJ1er as. 3, 2, 1
the s0J.u.t1on. of DE y 3

(D) 3x + 2y + z = c
3. 2 1 (C). 3x - 2y - z = c
z2 = c ( 8) -+ -+ -= C
(A) 3x2 + 2y2 + X y- Z (2)
dx _Q L.15 dy
. as l y z the sol utio n of DE z2
- 2yz. - •
y·,2 = y- + Z
= y- _ z ..... ..... . .
13. Using a set of multip lier I I

2 2·
z (·C) X+ y + z=C {D)· X + .) + t =C
(A) x2 + y2 + z2 = C (8) X+ 2+2 = C
v-

.......... ~ .,. . .. ..-: .. .,._.........:

•-
y

}:t: ···-.~•-· f ·--~ ~~ · ~ '~ . :~:~ i

} :;'/ ;:··:; ·~-,.: ~\.··:.t:\;~ ·"' 8. (C).
-1
6. (C). r 7. (A). r
5. (A).
~

3.. (A). 4. (D).

'-----
1. (BJ 2. (DJ
12. (D) 13. (B)
.__ 9. (C) 10. (D) 11. (B)
•••