Karnataka 2nd PUC Mathematics Differential Equations MCQ with Answers

GOVERNMENT OF KARNATAKA
DEPARTMENT OF SCHOOL EDUCATION (PRE-UNIVERSITY)
TH
18 CROSS, MALLESHWARAM, BENGALURU – 560 012
CHAPTERWISE MULTIPLE CHOICE QUESTIONS FOR COMPETATIVE EXAM
SUBJECT: MATHEMATICS – II PUC
NAME OF THE CHAPTER: DIFFERENTIAL EQUATIONS

Differential Equation: An equation containing an independent variable, dependent
variable and differential coefficient of dependent variable with respect to independent
variable is called a differential equation.

Eg:

Order of a Differential Equation: The order of a differential equation is the order of the
highest derivative appending in it.
Degree of a Differential Equation: The degree of a different equation is the degree of the
highest order derivative which occurs in it, after the differential equation has been made
free from radicals and fractions as for as the derivatives are concerned.
Solution of the Differential Equation: A relation between the variables without
derivatives which satisfy the given differential equation.
General solution: The solution which contains as many arbitrary constants as the order of
the differential equation is called general solution of the differential equa tion.
Particular Solution: Solution obtained by giving particular values to the arbitrary
constants in the general solution of the given equation.
Formation of Differential Equation: Let the given equation be ( )
Differentiating with respect to times we get more equations. By eliminating the
orbitanary constants we get the differential equation of order n.
Note: The number of independent arbitrary constants in the solution of a differential
equation is the order of the differential equation.
Variable Separable Differential Equation:
The differential equation is said to be in Variable separable form if is the
function of and n is the function of . The general solution is ∫ ∫ , where
is arbitrary constant.

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Homogeneous Differential Equation:
Homogeneous Function: A function ( ) is said to be homogeneous function of degree n if
( ) ( ) where n is a constant.
( )
General form of homogeneous differential e quation is ,
( )

by substituting , convert into variable separable form.

Eg:

Linear differential equations :

Type (i): General form ( ) ( ). Then ∫ ( ) is the integrating factor and the

solution is ∫ ( ) ∫ ( ) ∫ ( )

Type (ii): General form : ( ) ( ). Then ∫ ( ) is the integrating factor and the

solution is ∫ ( ) ∫ ( ) ∫ ( ) .

1. The Order and Degree of the differential equation ( ) is (Easy)

(a) 1, 2 (b) 2, 2 (c) 2,1 (d) 1, 1

2. The Order and Degree of the differential equation is (Average)

(a) 3,1 (b) 3,2 (c) 1,3 (d) 3, not defined

3. The Order and Degree of the differential equation ( ) is (Average)

(a) 4,1 (b) 4,2 (c) 2,4 (d) 4, not defined

4. The order and degree of the differential equation ( ) is (Easy)

(a) 1,4 (b) 2,1 (c) 2,4 (d) 4,2
5. The order and degree of the differential equation is (Easy)
(a) 1,2 (b) 2,1 (c) 2,2 (d) 2, not defined.

6. The order and degree of the differential equation ( ) ( ) ( ) is

(a) 2,3 (b) 3,2 (c) 3, not defined (d) 2, not defined

7. The order of the differential equation ( ) ( ) is (Easy)

(a) 3 (b) 2 (c) 1 (d) not defined.

8. The degree of the differential equation ( ) ( ) is (Average)

(a) 3 (b) 2 (c) 1 (d) not defined.
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9. The degree of the differential equation ( ) ( ) is (Easy)

(a) 3 (b) 2 (c) 1 (d) not defined.
10. Sum of the squares of the order and degree of a differential equation

( ) √ is [KCET 2026] (Difficult)
(a) 3 (b) 20 (c) 8 (d) 16
11. If ' ' and ' ' are the order and degree respectively of the differentiable equation.

( ) ( ) , then [KCET 2025] (Average)

(a) 1 (b) 2 (c) -1 (d) 0

12. The sum of the degree and order of the differential equation ( ) is
[KCET 2022] (Average)
(a) 4 (b) 6 (c) 5 (d) 7

13. The degree of the differential equation ( ) ( ) √

[KCET 2023] (Average)
(a) 1 (b) 2 (c) 2 (d) 6

14. The order of the differential equation ( ) =( ) is (Average)

(a) 6 (b) 3 (c) (d) 2

15. The degree of the differential equation ( ) is (Easy)

(a) 1 (b) (c) (d) not defined

16. W hat is the degree of the differential equation ( ) (Average)

(a) 1 (b) (c) (d) not defined
17. The sum of the degree and order of the differential equation

√ ( ) is (Average)

(a) 2 (b) (c) (d) not defined

18. The order & degree of the D.E. √ are (Average)

(a) 1 & 1 (b) 1 & 2 (c) 2 & 1 (d) 2 & 2

19. The degree of the differential equation ( ) ( ) ( ) is (Easy)

(a) 1 (b) (c) (d) not defined

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20. The sum of the degree and order of the differential equation [( ) ] is

(a) 2 (b) 3 (c) 4 (d) 5
5
 dy 2 d 2 y  4 d3y
21. The order and degree of the differential equation   +  = k (Average)
 dx  dx 
2
dx3

(a) 5,2 (b) 2,5 (c) 3,4 (d) 4,3
22. The order of the differential equation is always (Easy)
(a) Rational number (b) Whole number
(c) Negative integer (d) Positive integer

23. Find the number of arbitrary constants in the particular solution of a differential
equation of third order is: (Average)
(a) 3 (b) 2 (c) 1 (d) 0
24. The number of arbitrary constants in the general solution of a differential equation
of fourth order is (Easy)
(a) 4 (b) 3 (c) 2 (d) 1
25. The number of arbitrary constants in the solution of a differential equation of a
degree 3 and order 2 is (Easy)
(a) 3 (b) 2 (c) 5 (d) 1
26. The order of a differential equation whose general solution is y =A sinx + B cosx is
(Easy)
(a)4 (b) 2 (c) 0 (d) 3

27. If [ ( )] and f(0) = 0 then f(x) is equal to (Easy)

(a) (b) (c) (d)

28. The number of solutions of , when ( ) is [KCET 2021] (Easy)

(a) three (b) one (c) infinite (d) two
29. The equation y = mx +c is general solution of (Easy)

(a) x (b) y (c) (d)

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30. Solution of differential equation represents [KCET 2021] (Easy)
(a) A rectangular hyperbola. (b) Parabola whose vertex is at origin.
(c) Straight line passing through origin. (d) A circle whose centre is origin.
31. The family of curves represented by the differential equation is
(Easy)
(a) Parabolas (b) Circles (c) Ellipses (d) Hyperbolas

32. The solution of is (Easy)

a) c b) c) ( ) d) ( ) 0
ydx  xdy
33. The general solution of the differential equation  0 is (Average)
y
(a) xy = C (b) x = Cy2 (c) y = Cx (d) y = Cx2

34. If √ √ ( ) and , then [KCET 2026] (Average)

(a) 1 (b) (c) (d)

35. The solution of ( ) is [KCET 2024] (Average)
(a) (b)
(c) ( ) ( ) (d) ( ) ( )
36. The family of curves whose and intercepts of a tangent at any point are
respectively double the and coordinates of that point is [KCET 2024] (Average)
(a) (b) (c) (d)

37. The solution of ( ) which satisfy the condition y = 1 when x = 0.

(a) ( ) (b) ( ) (c) ( ) (d) ( )

38. The solution of the differential equation ( ) is [KCET 2022] (Difficult)

(a) ( ) (b) ( )
(c) ( ) (d) ( )

39. The solution of the differential equation ( ) is (Average)

(a) ( ) (b) ( ) (c) ( ) (d) ( )

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40. The solution of the differential equation is (Average)

(a) ( ) (b) ( )

(c) ( ) (d) ( )

41. The equation of the curve passing through the point (1, 1) such that the slope of
the tangent at any point ( ) is equal to the product of its co-ordinates is :
a) b) c) d)

42. A differential equation of the form = F (x, y) is said to be homogenous if

F(x, y) is a homogenous function of degree (Easy)
(a) 1 (b) 0 (c) 2 (d) 2, not defined
43. W hich of the following is a homogeneous differential equation? (Average)
(a) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0 (b) (xy) dx – (x3 + y3 )dy = 0
(c) (x3 + 2y2 ) dx + 2xy dy = 0 (d) y2 dx + (x2 – xy – y2 ) dy = 0

44. A homogeneous differential equation of the form ( ) can be solved by making

the substitution (Average)
(a) y = vx (b) v = yx (c) x = vy (d) x = v

45. The solution of ( ) ( ) is (Average)

(a) (b) (c) (d)

dy
46. Find the Integrating Factor of the differential equation x  2 y  x2  x  0 (Easy)
dx
(a) log x (b) x 2 (c) 2 log x (d) x

dy k
47. The Integrating Factor of the differential equation x - y = 2x 2 is , then k is(Easy)
dx x
(a) 1 (b) 1 (c) 2 (d) 2

48. The Integrating Factor of the differential equation ( ) is (Average)

1 1 1 1
(a) (b) (c) (d)
y 1
2
y2 1 1 y2 1 y2

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49. Integrating factor of the differential equation ( ) is

[KCET 2026] (Average)

(a) (b) ( ) (c) (d) √

50. The Integrating Factor of the differential equation dy + (y + 2x) dx = 0 is
then k is (Average)
(a) 1 (b) 1 (c) 2 (d) 2

51. General solution of the differential equation is

[KCET 2025] (Average)
(a) (b) (c) (d)

52. Integrating factor of the differential equation ( ) is (Average)

(a) (b) ( ) (c) (d) √
√

53. The integrating factor of the differential equation ( ) 2 is

(a) (b) log x (c) (d) x

54. Solution of the differential equation is (Average)

(a) (b) (c) (d)

55. The Integrating Factor of the differential equation is then k is

(a) 1 (b) 1 (c) 2 (d) 2
k
56. The Integrating Factor of the differential equation ( ) is ,
y
then k is (Average)
(a) 1 (b) 1 (c) 2 (d) 2

57. If , then ( ) ( ) [KCET 2022] (Average)

(a) (b) (c) (d)

58. General solution of the differential equation is (Average)

(a) (b) (c) (d)

59. The Integrating Factor of the differential equation is then k is

(Difficult)

(a) (b) (c) 2 (d) 2

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60. Statement 1: The Integrating factor of the differential equation ( ) is s

Statement 2: The Integrating factor of the differential equation is ∫ .

(a) Statement 1 is false and Statement 2 is true (Easy)
(b) Statement 1 is true and Statement 2 is false
(c) Statement 1 is true, statement 2 is true and statement 2 is a correct explanation for
statement 1
(d) Statement 1 is true, statement 2 is true and statement 2 is not a correct explanation
for statement 1

DIFFERENTIAL EQUATIONS
1 2 3 4 5 6 7 8 9 10
c d d b b d a d c b
11 12 13 14 15 16 17 18 19 20
d c d d d c b a d b
21 22 23 24 25 26 27 28 29 30
c d d a b b d b c c
31 32 33 34 35 36 37 38 39 40
b c c c d a c a a c
41 42 43 44 45 46 47 48 49 50
b b d c d b a d d a
51 52 53 54 55 56 57 58 59 60
a d b a c a b a b c

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