TRB BT Assistant - Ordinary Differential Equations Online Test (MCQ Quiz)

1. What is the order of the differential equation dy/dx + y = x?

A) 1 B) 2 C) 0 D) 3

2. What is the degree of the differential equation dy/dx + y = x?

A) 1 B) 2 C) 0 D) 3

3. Which of the following is a first-order ordinary differential equation?

A) d²y/dx² + y = 0 B) dy/dx + y² = x C) y'' - 3y' + 2y = sin(x) D) dy/dx + d²y/dx² = e^x

4. Which of the following is a linear ordinary differential equation?

A) y(dy/dx) + x = 0 B) dy/dx + y² = x C) dy/dx + xy = sin(x) D) d²y/dx² + y(dy/dx) = 0

5. The differential equation x(dy/dx) + y = x² is:

A) Non-linear and first-order B) Linear and first-order C) Non-linear and second-order D) Linear and second-order

6. What is the general solution of the differential equation dy/dx = 0?

A) y = cx B) y = c C) y = x + c D) y = cx²

7. What is the general solution of the differential equation dy/dx = k (where k is a constant)?

A) y = x + c B) y = cx C) y = kx + c D) y = c/k

8. The differential equation dy/dx = y/x is:

A) Exact B) Separable C) Homogeneous D) Linear

9. What is the integrating factor for the linear differential equation dy/dx + P(x)y = Q(x)?

A) ∫P(x)dx B) e^(∫P(x)dx) C) e^(-∫P(x)dx) D) ∫Q(x)dx

10. For the differential equation dy/dx + 2xy = x, what is the integrating factor?

A) e^(x²) B) e^(2x) C) e^(x²/2) D) x

11. What is the general solution of dy/dx = e^(x+y)?

A) e^(-y) = e^x + C B) e^y = e^x + C C) e^(-y) = -e^x + C D) e^y = -e^x + C

12. The differential equation M(x,y)dx + N(x,y)dy = 0 is exact if:

A) ∂M/∂y = ∂N/∂x B) ∂M/∂x = ∂N/∂y C) ∂M/∂y = -∂N/∂x D) ∂M/∂x = -∂N/∂y

13. Which of the following is a homogeneous differential equation?

A) dy/dx = x + y B) dy/dx = x² + y² C) dy/dx = x/y + y/x D) dy/dx = e^(x+y)

14. What is the degree of the differential equation (dy/dx)³ + x(dy/dx)² + y = 0?

A) 1 B) 2 C) 3 D) 0

15. What is the order of the differential equation y'' + 2y' + y = 0?

A) 0 B) 1 C) 2 D) 3

16. A differential equation is called 'autonomous' if:

A) It contains only the dependent variable and its derivatives. B) It contains only the independent variable and its derivatives. C) It does not explicitly depend on the independent variable. D) It is linear.

17. The differential equation dy/dx = f(x) is always:

A) Homogeneous B) Separable C) Exact D) Linear

18. What is the general solution of the differential equation dy/dx = 1/y?

A) y² = 2x + C B) y = x + C C) y² = x + C D) 2y = x + C

19. The differential equation dy/dx + P(x)y = 0 is:

A) Non-homogeneous linear B) Homogeneous linear C) Non-linear D) Second-order

20. What is the complementary function for the differential equation d²y/dx² - 5dy/dx + 6y = 0?

A) C₁e^(2x) + C₂e^(3x) B) C₁e^(-2x) + C₂e^(-3x) C) C₁e^(x) + C₂e^(6x) D) C₁e^(5x) + C₂e^(6x)

21. What is the particular integral for the differential equation d²y/dx² - y = e^(2x)?

A) e^(2x)/3 B) e^(2x)/5 C) 2e^(2x)/3 D) e^(2x)/(-3)

22. The differential equation y'' - 4y' + 4y = 0 has a general solution of the form:

A) C₁e^(2x) + C₂e^(-2x) B) (C₁ + C₂x)e^(2x) C) C₁e^(2x) + C₂xe^(2x) D) C₁e^(2x) + C₂xe^(-2x)

23. What is the integrating factor for the differential equation dy/dx + (1/x)y = x?

A) x B) x² C) 1/x D) ln(x)

24. The solution of the differential equation dy/dx = sin(x) with y(0) = 1 is:

A) y = -cos(x) + 2 B) y = cos(x) + 1 C) y = -cos(x) + 1 D) y = sin(x) + 1

25. Which type of differential equation is dy/dx = (x+y)/(x-y)?

A) Separable B) Linear C) Homogeneous D) Exact

26. What is the degree of the differential equation d²y/dx² + (dy/dx)² + y = 0?

A) 1 B) 2 C) 3 D) 0

27. The differential equation (2x + 3y)dx + (3x - y)dy = 0 is:

A) Exact B) Not exact, but can be made exact C) Homogeneous D) Separable

28. What is the integrating factor for x dy/dx - y = x²?

A) x B) 1/x C) x² D) 1/x²

29. The general solution of y'' + ω²y = 0 is:

A) C₁cos(ωx) + C₂sin(ωx) B) C₁e^(ωx) + C₂e^(-ωx) C) C₁ + C₂x D) C₁cos(ωx) - C₂sin(ωx)

30. What is the condition for the differential equation Mdx + Ndy = 0 to be homogeneous?

A) M and N are homogeneous functions of the same degree. B) M is homogeneous and N is constant. C) M and N are linear. D) M and N are of order 1.

31. If the roots of the auxiliary equation of a second-order linear homogeneous ODE are real and distinct, say m₁ and m₂, then the general solution is:

A) C₁e^(m₁x) + C₂e^(m₂x) B) (C₁ + C₂x)e^(m₁x) C) e^(αx)(C₁cos(βx) + C₂sin(βx)) D) C₁e^(m₁x) + C₂xe^(m₂x)

32. What is the particular integral of d²y/dx² - y = x?

A) -x B) -x - 1 C) x + 1 D) x

33. The differential equation dy/dx = (x+y+1)/(x+y+2) can be reduced to a separable equation by the substitution:

A) y = vx B) x = v C) u = x+y D) y = x+u

34. If the roots of the auxiliary equation are complex conjugates, α ± iβ, then the general solution is:

A) e^(αx)(C₁cos(βx) + C₂sin(βx)) B) C₁e^(αx) + C₂e^(iβx) C) C₁cos(αx) + C₂sin(βx) D) C₁e^(αx) + C₂e^(-αx)

35. What is the order of the differential equation (d³y/dx³) + x(d²y/dx²) + (dy/dx) + y = e^x?

A) 1 B) 2 C) 3 D) 0

36. The differential equation y'' + y = 0 is an example of:

A) A non-homogeneous linear ODE B) A homogeneous linear ODE with constant coefficients C) A non-linear ODE D) An autonomous ODE

37. What is the general solution of dy/dx = y?

A) y = Ce^x B) y = Ce^(-x) C) y = x + C D) y = Cx

38. The differential equation dy/dx = x²y is:

A) Linear and separable B) Non-linear and separable C) Linear and homogeneous D) Non-linear and homogeneous

39. What is the integrating factor for dy/dx - y = e^x?

A) e^x B) e^(-x) C) 1 D) e^(2x)

40. A differential equation is called separable if it can be written in the form:

A) M(x)dx + N(y)dy = 0 B) M(x)dx = N(y)dy C) f(x)dx = g(y)dy D) M(x,y)dx + N(x,y)dy = 0 where ∂M/∂y = ∂N/∂x

41. The solution to the initial value problem y' = -xy, y(0) = 1 is:

A) y = e^(-x²/2) B) y = e^(x²/2) C) y = e^(-x) D) y = e^x

42. Which differential equation is NOT homogeneous?

A) dy/dx = (x² + y²)/xy B) dy/dx = (x+y)/x C) dy/dx = y/x + sin(y/x) D) dy/dx = x + y + 1

43. What is the degree of the differential equation (d²y/dx²) + x(dy/dx) + y = 0?

A) 1 B) 2 C) 0 D) 3

44. The general solution of y'' - 2y' + y = 0 is:

A) C₁e^x + C₂xe^x B) (C₁ + C₂x)e^x C) C₁e^x + C₂e^(-x) D) C₁e^x + C₂xe^(-x)

45. The differential equation Mdx + Ndy = 0 is exact if there exists a function F(x,y) such that:

A) dF = Mdx + Ndy B) ∂F/∂x = M and ∂F/∂y = N C) ∂F/∂x = N and ∂F/∂y = M D) F(x,y) = C

46. What is the particular integral of d²y/dx² + y = sin(x)?

A) -x cos(x)/2 B) x cos(x)/2 C) -x sin(x)/2 D) x sin(x)/2

47. Which of the following is a Bernoulli differential equation?

A) dy/dx + y/x = x²y³ B) dy/dx + x²y = sin(x) C) dy/dx + y = 0 D) d²y/dx² + y = 0

48. To solve a Bernoulli equation dy/dx + P(x)y = Q(x)yⁿ, the standard substitution is:

A) u = y^(1-n) B) u = yⁿ C) u = y D) u = 1/y

49. The differential equation dy/dx = (x+y)² is separable after the substitution:

A) u = x+y B) y = vx C) u = x D) u = y

50. What is the order of the differential equation x²(d³y/dx³) + 2x(dy/dx) - y = 0?

A) 1 B) 2 C) 3 D) 0

Ordinary Differential Equations - Online Test

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