MCQ

1. 17. DIFFERENTIATION AND
ORDINARYDIFFERENTIAL
EQUATION
MULTIPLE CHOICE QUESTIONS
1]. Choose the correct statement from following,
- - -
(a) Every function has a limit
(b) The function having limit is continuous
(c) Every continuous function has limit
(d) The continuity implies differentiability
2. Apartial differential equation has,-
(a) One independent variable
(b) Two or more Independent variables
(c) More than one dependent variable
(d) Equal number ofdependent and independent variables
3J. Degree of differential equation is, - -
(a) Exponent power of highest derivative
(b) Highest derivative in the equation
(c) Lowest derivative in the equation
(d) Exponent power of lowest derivative
41. Orderofdifferentialequation is, -
(a) Lowest derivative in the equation
(b) Highest derivative in the equation
(c)Exponent power of highest derivative
(d) Exponent power of lowest derivative
114

2. and degrecofthedifferentialequation+ +y=x
J O r d e
are
respectively,
-
-
-
(b) 4,2
(a) 1, 2
(d)2,1
(c)
2,4
Theorderofthedifterential equation given by+4y = sinz
(b) 4
(d) 0
(a) 2
(c)1
1.
The steps to obtain a diferential equation are given below. Arrange
them in the correct order,
1) Using the equations obtained, eliminate the arbitrary constants.
2Differentiate the given function w,.r.t the independent variable
present in the equation.
3) Keep differentiating as many times as the number of arbitrary
constants.
Choose the correct order from the options given below.
(a)1,2,3
()2, 1,3
(b)
2,3,1
(d) 1,3, 2
81.Thesolution ofthesecond-order differential equation contains
arbitrary constants.
(a)1
c)3
(b) 2
(d) 0
91.Order and degree ofthe differential equation =0 are
respectively, - - -
(a) 1,2 (b) 3,2
() 2,2 (d)2,1
10).If F = 2x3y +2xy then Fx is, -
(a) 12x +6y
(c) 6xy + 4x
(b)6xy +4xy
(d) 12xy + 4y
1.If F =
3x3y +3xy3 then Fyy is, - - -
(a) 18xy
(c) 3x3 +9x
(b) 9xy
(d) 12x
115

3. ON
121. i s thecondition of- - differentiation.
dx
(a) P'artial (b) Successive
(c) Exact (d) Total
dy
131.Ify = cosx then 1,
(b) -2xsinx2
(d) 2xcosx2
(a) 2xcosx
(c) 2xsinx2
indas y=
fløtz)p
141. Which ofthe following gives chain rule to find
(a) X du (b) yd
( (c)=dx dx du dx
2
15]. Auxiliary equationofthe differentialequation-4
-4+2y=
canbe--
(a) m- 4m +4 =0
(c)m + 4m +4 = 0
(b)
m2 - 4m +2 =0
(d)m+4m +2=0
161. If F = 2x*y + 3xy - 4xy then Ey is---
a)6xy+6x-8
(c)6xy +6x - 8y
(b)6x +6x-8y
(d)6x +6xy -8
17. Which ofthe following is not a type of differential equation?
(a) Ordinary differential equation
(b) Successive differential equation
(c) Linear differential equation
(d) Homogeneous differential equation
18J. Which of the following is a second-order differential cquation?
(a)(y +x=y
(c)y'y"+y=sin x
191.Which ofthe following is the general solution dy
dx
(b) y'= y2
(d) y+x=y
dx
(a) y = (Ax + B)ex
(b) y = Acosx + Bsinx
(c) y = Ae* + Be-*
(d) y =(Ax + B)ex
116

4. 0The
se
(a)
y=e(r-1)
Thesolutionof+yey(0) =0is,,-
(b) y =
xe
(d)y =xe+1
(c)y = x e *
dy
Integrating factor ofthe differential equation+ytanx - secx =
dx
0is,
(b) cosx
(a) secx
(d) e secx
(c)e
os
21.Ify= e(Acosx + sinx), then y is a solution of,-.
6)+2=0
+ 2 y = 0
dy =
0
(a)+2+2y =
0
dx
1 Integrationfactorforthedifferentialequation x=1+2xy is, -.
(a)ex (b) e-2x
(c) ex< (d) e2x
241.1f the general solution of a differential equation is (y + c) = cx,
where c is an arbitrary constant, then the order and degree of
differential equation is, - -
(a) 1,2
(c)2,3
(b)2,1
(d) None of these
25].Integrationfactor forthe differential equation x-2xy=is--
dx
(a)ex (b) e2x
(d)-
261. Which of the following equations is an exact differential equation?
(a)(x+1)dx -xydy =0
(6) xdy + (3x -
2y)dx = 0
(c)xydy-ydx =0 (d)
2xydx + (2+r*)dy = 0
7Theequationy =
cx is general solution of---
(a)y'=
)y2
b)y=
(d)y'= 2x
117

5. the lincar differential
281.Which of the following is a solution of the lincar difta
cquation+ =x2?
(b)
xy=+c
(d)y =+c
(a)xy =+ 3
()xy=+c
29].The solution of a differential equation is y = Cqet* 4 Capz
differential equation is given by-
, the
a-7+ 12y = 0 (b)4-7+7y= 0
dx2
()+7+21y = 0 (d) None ofthese
dx
301. If p and q are the degree and order of the differential quation
(4)+3+=4,then thevalueof2p-3q is
2
dx dx
(b)-7
(a) 7
(c) 3
(d)-3
31.Thedegreeofthedifferential equation(1+ =
dx/
(b) 2
(d) 4
-
(a) 1
()3
32]. Which of the following is a solution of the differential equation
=ey +x3e-y ?
dx
(a)e =
e ++c (b)e =
ex ++c
(c)e =
ex
++c (d)e =
e++c
33.The differential equation 2ydx -
(3y -
2x)dy = 0 is, - - -
(a) Homogeneous and linear but not exact
(b) Exact and linear but not
homogeneous
(c) Exact, homogeneous and linear
(d) Exact and
homogeneous but not linear
341.Theorder of differential equation is
always, - -
(a) Rational number
(b) Whole Number
(c) Negative integer
(d) Positive integer
118

6. he differential equation 2cos(y) dx - xy sin(y )dy = 0 ha
(a)e
as an integrating factor
b) as an integrating factor
c)e as an integrating factor
d)3x as an integrating factor
LAn integration factor ofx+(3x +1)y =xe-lxis,--.
b) 3xe
(d)xe
( a )
x e 3 r
c)xe
2
71.Thedifferentialequation+ +r?+y' is,-..
(a)Homogeneous partial
(b) Non-homogeneous partial
(c) Non-homogeneous exact
(d) Homogeneous exact
38|. Particular integral of (D'-3D +2)y = e3* is,---
(
eSx
13
5x
()4
391. Which ofthe following is the linear diferential equation?
3
(k+6Hi
cos
()+ycosx = sinx
dx
(d) None ofthese
40.An auxiliary equation (A.E.) cannot have - - - roots.
(a) Real and unequal (distinct)
(6) Real and equal
(d) Complex and equal
(c) Complex conjugate
The higher-order partial derivatives i.e. second or third order are
called --partialdifferentials.
(a) Exact b) Total
(C) Successive (d) None ofthese
119

7. fferential
421.Which of the following correctly defincs ordinary
cquations?
variable (say
(a) A differential cquation in which a dependent variaht.
(say 'x)
y) depends on only one independent variable (
variable (say
(b) A differential cquation in which an independent variakl.
y) depends on only onc dependent variable (say 'x') y
(say
(c) A differential equation in whicha dependent variable(e
etc.)
depends on one or more independent variables (say'x',te
say
(d) A differetial equation in which an independent variablefa
y') depends on one or more dependent variables (say'x,t
of an
431. Which one of the following is not a criterion for linearit
ordinary differential equation?
(a) The dependent variable y and its derivatives are offirst
degree
(b) The derivatives of the dependent variable y should be of
second degree
(c) No product terms of y and/or any of its derivatives are present
(d) No transcendental functions ofy and/or its derivatives occur
441.Solutionofthedifferentialequationxy=1+x+y+xyis,..
dx
(a)(y-x)-log (x(1+y))= c
(b) (y +x)- log (x) = c
(clog (x(1 + y) = c
(d) y-«)-log (y(1 +«)=c
2
451.xy +yx* + =0 is a, - - - .
(a) Second order, third degree, and linear differential equation
(b) First order, third degree, linear differential equation
(c) Second order, third degree, and non-linear differential equation
(d) First order, third degree, non-linear differential equation
461. What is the order of the partial differential equation?
ou u = 0?
2u
Oxdy
(b) I
(a) 5
(c) 2 (d) 4
120

8. 17
A
mplementary
function of the differential cquation
481.The
roots
of
fauxiliary equatio. for the differential equation
-3+ 2x = 0 is,---
dt
(a)
Ce +Cze?t
( c ) C e + c 2 e t
dt
(b)cjet+ Ce-2t
(d) -3t e~2t
+2+5 0 are,- -
b) Real and unequal (distinct)
(d) None of these
(a) Real and equal
c Imaginary
491. A particular integral of the differential equation
5+6y = es* is, - - .
dr?
)e
(d);e*
(a)etx
(c)ex
S01. If A and B are constants and auxiliary equation has real and equal
roots m m2 = m, then general solution ofhomogeneous second
order differential equation is - - -
(a) (Ax -
B)e
mx
(c) (Ax + B)emx
(b) (Ax +B)e-2mx
(d) (Ax - B)emx
51.
The given differential equationt3+7y = sinx is,--
(a) Homogeneous
(C) Inhomogeneous
(b) Non-homogeneous
d) Noneofthese
52]. Ordinary differential equation contains,
(a) Only one variable
(6) Two or more independent variable
(c) Zero variables
(d) No independent varible
Iotal differential equation can be solving by homogeneous equation
method and, ---
(a) Inspection method
c) Reduction method
(b) Auxillary equation method
(d) All of these
121