Class 12 Math Ch-9 Differential Equations MCQs Exam 2027

Topic 1: Order and Degree of Differential Equations

1. The order of the differential equation $2x^2 \frac{d^2y}{dx^2} – 3 \frac{dy}{dx} + y = 0$ is: [BSEB, 2020 A]

   (A) 2

   (B) 1

   (C) 0

   (D) Not defined

2. The order and degree of the differential equation $xy \left( \frac{d^2y}{dx^2} \right) + x \left( \frac{dy}{dx} \right)^2 – y \frac{dy}{dx} = 0$ are: [BSEB, 2024 A]

   (A) order = 2, degree = 1

   (B) order = 2, degree = 2

   (C) order = 1, degree = 2

   (D) order = 1, degree = 1

3. The order and degree of the differential equation $1 + \left( \frac{dy}{dx} \right)^2 = \left( \frac{d^2y}{dx^2} \right)^3$ are: [BSEB, 2018 A]

   (A) order = 2, degree = 3

   (B) order = 1, degree = 2

   (C) order = 2, degree = 2

   (D) None of these

4. The order of the differential equation $\frac{dy}{dx} + y = e^x$ is: [BSEB, 2017 C]

   (A) 2

   (B) -1

   (C) 1

   (D) -2

5. The order of the differential equation $\frac{d^2y}{dx^2} + x^3 \left( \frac{dy}{dx} \right)^3 = x^4$ is: [BSEB, 2016 C, 2019 A]

   (A) 1

   (B) 2

   (C) 4

   (D) 3

6. The degree of the differential equation $1 + \left( \frac{dy}{dx} \right)^2 + \frac{d^2y}{dx^2}$ is: [BSEB, 2016 C]

   (A) 1

   (B) 2

   (C) 3

   (D) 0

7. The degree of the equation $\left( \frac{d^2y}{dx^2} \right)^2 – x\left( \frac{dy}{dx} \right)^3 = y^3$ is: [BSEB, 2016 A]

   (A) 0

   (B) 1

   (C) 2

   (D) 3

8. The order of the differential equation $\left( \frac{d^2y}{dx^2} \right)^3 + 2 \left( \frac{dy}{dx} \right)^3 + 9y = \sin x$ is: [BSEB, 2021 A]

   (A) 3

   (B) 4

   (C) 2

   (D) None of these

9. The order of the differential equation $\frac{d^2y}{dx^2} + \left( \frac{dy}{dx} \right)^3 + y = 0$ is: [BSEB, 2015 A]

   (A) 1

   (B) 2

   (C) 3

   (D) 0

10. The degree of the differential equation $\frac{dy}{dx} – \cos x = 0$ is: [BSEB, 2024 A]

    (A) 0

    (B) 1

    (C) 2

    (D) Not defined

11. The order of the differential equation $\left( \frac{dy}{dx} \right)^2 + y = x$ is: [BSEB, 2016 A]

    (A) 0

    (B) 1

    (C) 2

    (D) 3

12. The degree of the differential equation $\frac{d^2y}{dx^2} + y = 0$ is: [BSEB, 2017 A]

    (A) 1

    (B) 2

    (C) 0

    (D) 3

13. The degree of the differential equation $\left( \frac{d^2y}{dx^2} \right)^2 + \sin \left( \frac{dy}{dx} \right) + y = 0$ is: [BSEB, 2021 A]

    (A) 2

    (B) 1

    (C) 0

    (D) Not defined

14. The degree of the equation $\left( \frac{dy}{dx} \right)^4 + 3y \left( \frac{d^2y}{dx^2} \right) = 0$ is: [BSEB, 2017 C]

    (A) 4

    (B) 1

    (C) 2

    (D) 3

15. The degree of the differential equation $\left( \frac{dy}{dx} \right)^2 – y = x$ is: [BSEB, 2016 A]

    (A) 1

    (B) 2

    (C) 0

    (D) Not defined

16. The order of the differential equation $xy \frac{d^2y}{dx^2} + x \left( \frac{dy}{dx} \right)^2 – y \frac{dy}{dx} = 0$ is: [BSEB, 2024 A]

    (A) 3

    (B) 1

    (C) 0

    (D) 2

17. The order of the differential equation $\frac{dy}{dx} + 4y = \cos x$ is: [BSEB, 2022 A]

    (A) 0

    (B) 1

    (C) 2

    (D) Not defined

18. The degree of the differential equation $(\frac{d^2y}{dx^2})^2 + (\frac{dy}{dx})^3 + \sin y = 0$ is: [BSEB, 2024 A]

    (A) 1

    (B) 2

    (C) 3

    (D) Not defined

19. The degree of the differential equation $\frac{dy}{dx} – y \cos x = 0$ is: [BSEB, 2019 A]

    (A) 0

    (B) 1

    (C) 2

    (D) Not defined

Topic 2: Linear Differential Equations & Integrating Factor (I.F.)

20. The integrating factor of the differential equation $\frac{dy}{dx} + 2y = \sin x$ is: [BSEB, 2024 A]

    (A) $e^x$

    (B) $e^{3x}$

    (C) $e^{2x}$

    (D) $e^{4x}$

21. The solution of a linear differential equation of the form $\frac{dy}{dx} + P y = Q$ is: [BSEB, 2021 A]

    (A) $y \cdot e^{\int P dx} = \int (Q \cdot e^{\int P dx}) dx + C$

    (B) $y \cdot e^{\int Q dx} = \int (P \cdot e^{\int Q dx}) dx + C$

    (C) $x \cdot e^{\int P dy} = \int (Q \cdot e^{\int P dy}) dy + C$

    (D) None of these

22. The integrating factor of the equation $\frac{dx}{dy} + Px = Q$ is: [BSEB, 2018 C]

    (A) $e^{\int Pdx}$

    (B) $e^{\int Pdy}$

    (C) $e^{\int Qdx}$

    (D) $e^{\int Qdy}$

23. The integrating factor of the differential equation $\frac{dy}{dx} + y \tan x = \sec x$ is: [BSEB, 2021 A]

    (A) $\cos x$

    (B) $\sec x$

    (C) $\tan x$

    (D) $\sin x$

24. The integrating factor (I.F.) of the differential equation $\frac{dy}{dx} + \frac{y}{x} = e^x$ is: [BSEB, 2021 A]

    (A) $x$

    (B) $\log x$

    (C) $e^x$

    (D) $\frac{1}{x}$

25. The integrating factor of the differential equation $\cos^2 x \frac{dy}{dx} + y = \tan x$ is: [BSEB, 2020 A]

    (A) $e^{\tan x}$

    (B) $e^{\cot x}$

    (C) $e^{\sin x}$

    (D) $e^{\cos x}$

26. The integrating factor of the differential equation $\frac{dy}{dx} + \frac{2}{x} y = x$ is: [BSEB, 2019 A]

    (A) $x$

    (B) $x^2$

    (C) $e^x$

    (D) $\log x$

27. The integrating factor of the differential equation $\frac{dy}{dx} – y \sin x = \cot x$ is: [BSEB, 2023 A]

    (A) $\sin x$

    (B) $e^{-\sin x}$

    (C) $e^{\sin x}$

    (D) $e^{\cos x}$

28. The integrating factor of the differential equation $\frac{dy}{dx} + 2y = e^{3x}$ is: [BSEB, 2024 A]

    (A) $e^{3x}$

    (B) $e^{2x}$

    (C) $e^{x}$

    (D) $e^{4x}$

29. The integrating factor of the linear differential equation $\frac{dy}{dx} + Py = Q$ is: [BSEB, 2021 A]

    (A) $\int Pdy$

    (B) $\int Qdx$

    (C) $e^{\int Qdy}$

    (D) $e^{\int Pdx}$

30. The integrating factor of the differential equation $\frac{dy}{dx} – y \cos x = \sin x \cos x$ is: [BSEB, 2019 A]

    (A) $e^{-\sin x}$

    (B) $e^{\sin x}$

    (C) $e^{-\cos x}$

    (D) $e^{\cos x}$

31. The integrating factor of the differential equation $\frac{dy}{dx} + y \sec x = \tan x$ is: [BSEB, 2019 A]

    (A) $\sec x + \tan x$

    (B) $\sec x – \tan x$

    (C) $\sec x$

    (D) $\tan x \sec x$

32. The integrating factor of the differential equation $\frac{dy}{dx} + \frac{2y}{x} = 5x^2$ is: [BSEB, 2023 A]

    (A) $\frac{2}{x}$

    (B) $2e^x$

    (C) $2\log x$

    (D) $x^2$

33. The integrating factor (I.F.) of the differential equation $x \frac{dy}{dx} – y = 2x^2$ is: [BSEB, 2015 A]

    (A) $e^{-x}$

    (B) $e^x$

    (C) $\frac{1}{x}$

    (D) $x$

34. The integrating factor of the differential equation $\frac{dy}{dx} + 2y = \cos x$ is: [BSEB, 2022 A]

    (A) $e^{2x}$

    (B) $e^{-2x}$

    (C) $e^{\cos x}$

    (D) None of these

35. The integrating factor of the linear differential equation $\frac{dy}{dx} + y \cot x = 2 \cos x$ is: [BSEB, 2021 A]

    (A) $\sin x$

    (B) $\cos x$

    (C) $\cot x$

    (D) $e^{\sin x}$

36. The integrating factor of the differential equation $\frac{dy}{dx} + \frac{y}{x} = \frac{y^2}{x^2}$ is: [BSEB, 2018 A]

    (A) $\log x$

    (B) $x$

    (C) $\frac{1}{x}$

    (D) None of these

37. The integrating factor (I.F.) of the equation $\frac{dy}{dx} + Py = Q$ is: [BSEB, 2021 A]

    (A) $\int P \, dx$

    (B) $e^{\int P \, dx}$

    (C) $e^{\int Q \, dx}$

    (D) $\int Q \, dx$

38. The integrating factor of the linear differential equation $\frac{dy}{dx} + \frac{y}{x} = x^3$ is: [BSEB, 2019 C]

    (A) $e^x$

    (B) $\log x$

    (C) $x$

    (D) None of these

Topic 3: General and Particular Solutions of Differential Equations

39. The general solution of the differential equation $e^x dx + e^y dy = 0$ is: [BSEB, 2024 A]

    (A) $e^x + e^y = C$

    (B) $e^{x-y} = C$

    (C) $e^{x+y} = C$

    (D) $e^x – e^y = C$

40. The solution of the differential equation $xdx + ydy + \frac{xdy – ydx}{x^2 + y^2} = 0$ is: [BSEB, 2018 C]

    (A) $\frac{x^2}{2} + \frac{y^2}{2} + \log(x^2 + y^2) + c$

    (B) $\frac{x^2}{2} + \frac{y^2}{2} + \log(x^2 – y^2) + c$

    (C) $\frac{x^2}{2} + \frac{y^2}{2} + \tan^{-1} \frac{x}{y} + c$

    (D) $\frac{x^2}{2} + \frac{y^2}{2} + \tan^{-1} \frac{y}{x} + c$

41. The solution of the differential equation $\frac{dy}{dx} = \frac{1+y^2}{1+x^2}$ is: [BSEB, 2020 A]

    (A) $\tan^{-1} y + \tan^{-1} x = C$

    (B) $\tan^{-1} y – \tan^{-1} x = C$

    (C) $\tan^{-1} y \cdot \tan^{-1} x = C$

    (D) None of these

42. The general solution of the differential equation $\frac{dy}{dx} = e^{x+y}$ is: [BSEB, 2024 A]

    (A) $e^x + e^{-y} = K$

    (B) $e^x + e^y = K$

    (C) $e^{-x} + e^y = K$

    (D) $e^{-x} + e^{-y} = K$

43. The solution of the differential equation $e^{3x}dx + e^{4y}dy = 0$ is: [BSEB, 2023 A]

    (A) $e^{3x+4y} = k$

    (B) $e^{3x} + e^{4y} = k$

    (C) $\frac{1}{3}e^{3x} + \frac{1}{4}e^{4y} = k$

    (D) $e^{3x} + e^{4y} + e^{3x+4y} = k$

44. The solution of the differential equation $\frac{dx}{x} + \frac{dy}{y} = 0$ is: [BSEB, 2023 A]

    (A) $xy = k$

    (B) $\frac{x}{y} = k$

    (C) $x + y = k$

    (D) $x – y = k$

45. The solution of the differential equation $\frac{dy}{dx} + y = 1, y \neq 1$ is: [BSEB, 2022 A]

    (A) $y = 1 + Ae^{-x}$

    (B) $y = Ae^x$

    (C) $y = Ae^{2x} + 1$

    (D) $y = 1 + Ae^{3x}$

46. The solution of the differential equation $xdx + ydy = 0$ is: [BSEB, 2019 C]

    (A) $x^2 + y^2 = c$

    (B) $x^2 – y^2 = c$

    (C) $x + y = c$

    (D) None of these

47. The solution of the differential equation $xdy + ydx = 0$ is: [BSEB, 2022 A]

    (A) $\frac{x^2}{2} + \frac{y^2}{2} = k$

    (B) $\frac{x^2}{2} – \frac{y^2}{2} = k$

    (C) $xy = k$

    (D) None of these

48. The general solution of the differential equation $\frac{dy}{dx} = \frac{x}{y}$ is: [BSEB, 2022 A]

    (A) $x^2 – y^2 = k$

    (B) $y^2 – x^2 = k$

    (C) $x^2 + y^2 = k$

    (D) $xy = k$

49. The solution of the differential equation $(x + 4) (dx – dy) = dx + dy$ is: [BSEB, 2016 C]

    (A) $x – y = \log(x+y) + C$

    (B) $x + y = \log(x-y) + C$

    (C) $x^2 + y^2 = x + y + C$

    (D) $x^2 – y^2 = x + y + C$

50. Which of the following is a homogeneous differential equation? [BSEB, 2019 A]

    (A) $x^2 y dx – (x^3 + y^3) dy = 0$

    (B) $(xy) dx – (x^4 + y^4) dy = 0$

    (C) $(2x + y – 3) dy – (x + 2y – 3) dx = 0$

    (D) $(x – y) dy = (x^2 + y + 1) dx$

51. The solution of the differential equation $dx + dy = 0$ is: [BSEB, 2023 A]

    (A) $x = ky$

    (B) $x^2 + y^2 = k$

    (C) $x + y = k$

    (D) $xy = k$

52. The solution of the differential equation $x^2 dx + y^2 dy = 0$ is: [BSEB, 2023 A]

    (A) $x^3 + y^3 = k$

    (B) $x^2 + y^2 = k$

    (C) $x^2 – y^2 = k$

    (D) $x^3 + y^3 = 3k$

53. The solution of the differential equation $\frac{dy}{dx} = \frac{1+y}{1+x}$ is: [BSEB, 2020 A]

    (A) $(1+x)(1+y) = k$

    (B) $(1+y) = k(1+x)$

    (C) $y = x + k$

    (D) $1+x+y = k$

54. The solution of the differential equation $ydx – xdy = xydx$ is: [BSEB, 2018 A]

    (A) $\frac{y^2}{2} – \frac{x^2}{2} = xy + c$

    (B) $x = kye^x$

    (C) $x = kye^y$

    (D) None of these

55. The solution of the differential equation $\frac{dy}{dx} = \frac{y}{x}$ is: [BSEB, 2024 A]

    (A) $y = \log |x| + c$

    (B) $y = cx$

    (C) $y = x \log |x| + cx$

    (D) $y = \log |x| + cx$

56. The solution of the differential equation $(x + y) (dx – dy) = dx + dy$ is: [BSEB, 2018 C]

    (A) $\frac{x^2}{2} – \frac{y^2}{2} = x + y + c$

    (B) $x – y = \log(x + y) + c$

    (C) $\frac{x^2}{2} – \frac{y^2}{2} + xy = x + y + c$

    (D) $x + y = \log(x – y) + c$

57. The solution of the differential equation $\frac{dy}{dx} = e^{x-y}$ is: [BSEB, 2022 A]

    (A) $e^x + e^y = C$

    (B) $e^x – e^y = C$

    (C) $e^x \cdot e^y = C$

    (D) $e^{x+y} = C$

58. The solution of the differential equation $\frac{ydx – xdy}{y^2} = 0$ is: [BSEB, 2021 A]

    (A) $\frac{y}{x} = k$

    (B) $\frac{x}{y} = k$

    (C) $\frac{x}{y^2} = k$

    (D) None of these

59. The solution of $(1 + x^2) dy = (1 + y^2) dx$ is: [BSEB, 2018 C]

    (A) $x + y = k(1 – xy)$

    (B) $y – x = k(1 – xy)$

    (C) $x + y = k(1 + xy)$

    (D) $y – x = k(1 + xy)$

60. The solution of the differential equation $xdx + \frac{xdy – ydx}{x^2 + y^2} = 0$ is: [BSEB, 2018 A]

    (A) $\frac{x^2}{2} + \tan^{-1} \frac{x}{y} = k$

    (B) $\frac{x^2}{2} + \tan^{-1} \frac{y}{x} = k$

    (C) $\frac{x^2}{2} – \tan^{-1} \frac{y}{x} = k$

    (D) $\frac{x^2}{2} – \tan^{-1} \frac{x}{y} = k$

61. What is the particular solution of the differential equation $x dy + y dx = 0$ if $y(1) = 1$: [BSEB, 2023 A]

    (A) $xy = 1$

    (B) $x + y = 2$

    (C) $x^2 + y^2 = 2$

    (D) $\log(xy) = 1$