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MCQ (MTH108) ( Units -1,II, III)
1.The general solution of the differential equation
(a)
(b)
(c)
(d)
2. What is the wronskian of
is
(b)
(c)
(d)
3. If differential equation
is normal over the interval I then I is
(a)
(b)
(c)
(d)
4. If
then by the method of undetermined coefficient the assumed particular
integral
(a)
(b)
(c)
(d)
5. Particular integral for the differential equation
is
(a)
(b)
(c)
(d)
6. The particular integral of the differential equation
(a)
(b)
(c)
(d)
7. The general solution of the differential equation
(a)
if
is
is
(b)
(c)
(d)
8. If
is
(a)
(b)
(c)
(d)
where
then the characteristic equation of A
9. For the system of the differential equations
, the value of
is given by
(a)
(b)
(c)
(d)
10. If
is solution of
then
a) 1
is
b) 2
c)
d)
11. what is lowest order of differential equation whose particular integral is
a) 2
b) 3
c) 4
12. If system of equations
then its eigen values are
a) -2, -2
d) 6
be solved by matrix method
b) -5, 1
c) -1, -3
13. If system of equations
d) -6, 2
, has eigen values 1 and
then
which of following is not Eigen vectors corresponding to value 1
a)
14.
b)
If
, b , c are functions of x and
formula is
a)
15.
c)
be second order differential equation where a
on any interval then wronskian according to Abel’s
b)
c)
If
roots are real and distinct if
a)
d) None.
be second order differential equation then its
b)
c)
16.
d)
d) both a and b
be solution of
a)
b)
17. If
c)
d)
then
a)
b)
c)
18. Which of the following is true?
a)
b)
c)
d)
d)
19. If the system of non-homogeneous simultaneous equations are given by
where
and A=
and
then the trial solution for the
particular integral of above system is given by (Where
a)
above
b)
)
c)
d) None of
20. If
satisfy the equations
=0,
=0 and
is the
second order differential equation satisfied by
then what is the value of b
a) 5
b)
c) 3
d)
21. When the given system of linear differential
equation
form
where
are expressed into a matrix
then what is the value of A
a)
(b)
(c)
22. For a given system of linear differential equation
second order linear differential satisfied by the
is
a)
b)
(d)none
(d)
, the
c)
23. The general solution of system of simultaneous equations given by
and A=
a)
values.
b)
c)
d)
, where
is
, where
, where
, where
, where
are corresponding vectors of eigen
are corresponding vectors of eigen values
are corresponding vectors of eigen values
are corresponding vectors of eigen values
24. What are the characteristic roots of a homogeneous LDE having 4 + x e2x as its
particular solution?
a) 0, 2
b) 4, 2
c) 4, 2 ,2
d) 0, 2, 2
25. The general solution of y(4) - 2y’’’ = 0 is given by
a) A + B e2x
b) A + B + Ce2x
c) A + Bx + Cx2 + De2x
2x
+ Dxe
where A, B, C and D are arbitrary constants.
d) A + Bx + Cx2
26. The DE x2y’’ – 4xy’ +6y = 0 on (0, ∞) has ___ linearly independent solutions.
a) 2
b) 3
c) infinite
d) Can’t say
27. The general solution of the equation 4y’’ - 4y’ + y = 8ex/2 is given by
a) Aex/2 + Bex/2 + x2ex/2
b) Aex/2 + Bxex/2 + x2ex/2
c) Aex/2 + Bex/2 + ex/2
d) Aex/2 + Bxex/2 + xex/2
28. If
, then what will be the Complementary Factor?
a) C1e  x  C1e 2 x
b) C1e x  C1e 2 x
c) C1e

x
3
 C1e

2x
3

4x
e2 x e 7
d)

5
7
d2 y
dy
29. If
 P  Qy  eaxV ,V is a function of x , then Particular Integral will be written as
2
dx
dx
1
1
1
1
a)
b) e ax
c) eax
V
V
V d) eax
V
2
f ( D  a)
f ( D  a)
f ( D  a)
f (D  a2 )
30. The auxiliary equation of
a)
b)
c)
d)
31. The auxiliary equation of the second order Euler Cauchy differential equation
has roots
a) -3, 2
b) 3, 2
c) -3 ± 2i
d) 3 ± 2i
32. The characteristic equation of
solution?
a) (A + B lnx) x1/2
b) (A + Bx) ex/2
c) Ax1/2
d) Aex/2
has double root ½, what is its general
33.If m-1 and m+2 are factors of auxiliary equation of y   y   2 y  0 then general solution
is
34.
35.
(a)
(c)
Ae  x  Be 2 x (b)
e x  e 2 x
Ae x  Be 2 x (d)
ex  e2x
The Wronskian of the functions f(x)=secx and g(x)=tanx is
(a)
1
(b)
-1
(c)
secx
(d)
tanx
x
2x



If e
and e are solutions of y  y  6 y  0 then roots of auxiliary equation are
(a) 1 and -2 (b) -1 and 2
(c) 1 and 2
(d)
-1 and -2
36.
The general solution of 4 y   9 y   2 y  0 is
37.
(a) Ae 2 x
(b) Ae  Be 4
(c)  A  Bx e
The general solution of 4 y   4 y   17 y  0 is
x
2x
x
2
Ae
(a)
2x 
2x
(d)
 Be 2 x (b) Ae2 x  Be 2 x
x
x
(c) e  A cos  B sin 
2
2

(d)
x
2
e
 A cos 2 x  B sin 2 x 
 A  B e 2 x
38.
The general solution of y   4 y   5 y   2 y  0 is
(a) Ae
2 x
 B  Cx e  x
2 x
 B  C xe x
(c) Ae
39.
41.
42.
43.
(d)
If D is a differential operator then value of
2 x
Ae  x  B  Cx e 2 x
 A  Bx e  x  C  Dx e 2 x
1 2 x
(e  Sm2 x  4)
D
2 x
2 x
Cos2 x
Cos2 x
Cos2 x
 4x
 4x
(b)  e 
(c)  e 
2
2
2
2
2
2
2 x
Cos 2 x
e

4
(d) 
2
2
2x
Particular Integral of y   2 y   3 y  e is
1
1 2x
1
1
 e2 x

(a)
(b)
(c)
(d)
e
5
5
5
5
Particular Integral of y   4 y  Sin 2 x is
1
x
x
(a)
(b)
(c)
(d)
 cos 2 x
cos 2 x
 cos 2 x
2
2
2
 x cos 2 x
1
If D is a linear differential operator then
.e x 
f ( D)
1
.e x
2
1
f ( )
(a)
, f ( 2 )  0
(b)
.e x
f (D   )
1
.e x
2
1
f ( )
(c)
, f ( 2 )  0 (d)
, f ( )  0
.e x
f ( )
(a)  e
40.
(b)

In method of undetermined coefficients if complimentary function
yc  Ae x  B  Cxe 2 x of equation y   3 y   4 y  e 2 x then choice of particular
integral will be
2x
44.
45.
(a) cxe
(b) cx 2 e 2 x
(c) ce 2 x
(d) c1e  x  C2 e 2 x
In method of undetermined coefficients. If complementary function
yc  ACos2 x  BSin 2 x of equation y   4 y  Sin 2 x then choice of particular integral
(a)
c1 xCos2x+ c2Sin2x
(b)
c1Cos2x+ c2xSin2x
(c)
x[c1Cos2x+ c2Sin2x]
(d)
c1Cos2x+ c2Sin2x
2


General solution of x y  xy  4 y  0
y  Ax  Bx 2 (b) y  Ax 2  Bx 2
y  Ax 2  Bx 2
(a)
46.
2
(c) y  Ax  Bx (d)
2


General solution of x y  3xy  10 y  0
(a)
(c)
yx
1
y  xACos3log x  BSin 3log x (b) y  xACoslog x  BSin log x
y  x 1 ACoslog x   BSin log x 
ACos3 log x   BSin 3 log x 
(d)
47.
General solution of system of simultaneous equations
y1  2 y1  y2
y2  y1  2y2
48.
(a)
y1  Ae t  Be 3t , y 2  Ae t  Be 3t (b) y1  Ae t  Be 3t , y 2  Ae t  Be 3t
(c)
y1  Ae t  Be 3t , y 2  Ae t  Be 3t (d) y1  Ae t  Be 3t , y 2  Ae t  Be 3t
General solution of system of simultaneous equations
y1  y1  2 y 2  6t 2  12t  9
(a)
(b)
(c)
(d)
y 2  4 y1  3 y 2  4t 2  3t  6
2
27
21 121
y1  Ae t  Be 5t  2t 2  t 
, y 2   Ae t  2 Be 5t  4t 2  t 
5
25
5
25
2
27
21
121
y1  Ae t  Be 5t  2t 2  t 
, y 2   Ae t  2 Be 5t  4t 2  t 
5
25
5
25
2
27
21 121
y1  Ae t  Be 5t  2t 2  t 
, y 2   Ae t  2 Be 5t  4t 2  t 
5
25
5
25
2
27
21 121
y1  Ae t  Be 5t  2t 2  t 
, y 2   Ae t  2 Be 5t  4t 2  t 
5
25
5
25
49.The equation y' ' xy'6 y  ln( x 2  9) is normal in any of the subintervals of
a) (3, ) b) (0, ) c) (0,4) d) (,3)  (3, )
50. A2x+B(6x+3)+C(3x+2)=0 will be linearly independent when
a) A=B=C b) A=B=C=0 c) Wronskian(A,B,C)=0 d) A=B=C=1
51. General solution of
a)
b)
52. General solution of
a)
b)
is
c)
d)
is given by
c)
d)
53. Particular integral for y' '16 y  cos4x is
a) 2 x sin 4 x b) 
x sin 4 x
x sin 4 x
c)
d) None of these
8
8
54. The diff. equation of ( D 2  6D  9) y  50e 2 x has P.I.
2 2x
a) e
b) 2e 2 x c) e 2 x d) None of these
3
55.By method of undetermined coefficient , the choice of particular integral for y' '4 y  5e 2 x is
a) C e 2 x
b) C xe 2 x
c) C x 2 e 2 x d) C x 3 e 2 x
56. By variation of parameters y' '4 y  cos2x ,the value of wronskian is
a)1 b) 2 c) 4 d) none of these
57. General solution of y' '16 y  12e2 x is
3
5
3
5
a) Ae 4 x  Be4 x  e 2 x b) A cos 4 x  B sin 4 x  e  2 x c) Ae 4 x  Bxe4 x  e2 x d) none of these
58. The solution of diff. equation x 2 y' ' xy' y  0 is
a) c1 cos(ln x)  c2 sin(ln x) b) c1 x  c2 x 2 c) c1 cos x  c2 sin x d) None of these
59.By elimination, solution of
a)
b)
in the simultaneous system
will be
c) 0 d) Does not exist
60.
has auxiliary equation
a) m2+2m-4=0 b) m2-4=0 c) m2-2m-4=0 d) m2+4=0
61. The roots of auxiliary equation of x 2 y' ' xy'9 y  0 are
a)3,3 b) 3,3 c) 3, 3 d)
62. By matrix method , Eigen values for
will be
a) 6,1 b) -6,-1 c) 2,3 d) -3, -2
NOTE- Take more MCQ questions from reference book B.S.Grewal 43rd
edition page no. 500-501