TRB BT Assistant - Homogeneous, Exact and Linear Equations Online Test (MCQ Quiz)

1. What is the general form of a first-order homogeneous differential equation?

A) dy/dx = f(y/x) B) M(x, y)dx + N(x, y)dy = 0 where M and N are homogeneous of the same degree C) dy/dx + P(x)y = Q(x) D) ∂M/∂y = ∂N/∂x

2. Which substitution is typically used to solve a homogeneous differential equation dy/dx = f(y/x)?

A) y = vx B) x = vy C) y = v D) x = v

3. If a differential equation is written as M(x, y)dx + N(x, y)dy = 0, what condition makes it an exact differential equation?

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

4. For an exact differential equation M(x, y)dx + N(x, y)dy = 0, the solution is given by F(x, y) = C, where ∂F/∂x = M and which of the following?

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

5. What is the integrating factor for a first-order linear differential equation of the form dy/dx + P(x)y = Q(x)?

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

6. When solving a linear differential equation dy/dx + P(x)y = Q(x), after multiplying by the integrating factor, the left side becomes the derivative of what expression?

A) y * (integrating factor) B) x * (integrating factor) C) y + (integrating factor) D) y / (integrating factor)

7. Consider the differential equation (x^2 + y^2)dx + 2xy dy = 0. Is this equation homogeneous?

A) Yes, M and N are homogeneous of degree 2 B) No, M is degree 2 but N is degree 3 C) Yes, M is degree 2 but N is degree 1 D) No, it is not homogeneous

8. What is the correct substitution for solving the homogeneous equation dy/dx = (x^2 + y^2) / (xy)?

A) y = vx B) x = vy C) y = v D) x = v

9. For the differential equation (2xy)dx + (x^2 + y^2)dy = 0, check if it is exact. Here M(x, y) = 2xy and N(x, y) = x^2 + y^2.

A) Not exact, because ∂M/∂y = 2x and ∂N/∂x = 2x B) Exact, because ∂M/∂y = 2x and ∂N/∂x = 2x C) Not exact, because ∂M/∂y = 2y and ∂N/∂x = 2x D) Exact, because ∂M/∂y = 2y and ∂N/∂x = 2y

10. If an equation is exact, M(x, y)dx + N(x, y)dy = 0, and we find a potential function F(x, y) such that ∂F/∂x = M, what is the next step to find F?

A) Integrate M with respect to x, treating y as a constant B) Integrate N with respect to y, treating x as a constant C) Integrate M with respect to y, treating x as a constant D) Integrate N with respect to x, treating y as a constant

11. Consider the linear differential equation dy/dx + (2/x)y = x. What is P(x)?

A) 2/x B) x C) -2/x D) 1

12. Calculate the integrating factor for dy/dx + (1/x)y = x^2.

A) x B) x^2 C) 1/x D) ln(x)

13. The equation dy/dx = y/x + tan(y/x) is an example of which type of differential equation?

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

14. Solve the homogeneous equation dy/dx = y/x. (Hint: y=vx)

A) y = Cx B) y = Cx^2 C) y = C/x D) y = C

15. For the equation (e^x sin y)dx + (e^x cos y)dy = 0, what is ∂M/∂y and ∂N/∂x?

A) ∂M/∂y = e^x cos y, ∂N/∂x = e^x cos y B) ∂M/∂y = e^x cos y, ∂N/∂x = e^y cos x C) ∂M/∂y = e^x sin y, ∂N/∂x = e^x cos y D) ∂M/∂y = e^y cos x, ∂N/∂x = e^x cos y

16. Since (e^x sin y)dx + (e^x cos y)dy = 0 is exact, find the potential function F(x, y) by integrating M = e^x sin y with respect to x.

A) e^x sin y + g(y) B) e^x cos y + h(x) C) e^y sin x + g(y) D) e^y cos x + h(x)

17. To find the constant of integration g(y) in the potential function F(x, y) = e^x sin y + g(y), we differentiate F with respect to y and equate it to N.

A) ∂F/∂y = e^x cos y + g'(y) = N B) ∂F/∂y = e^x sin y + g'(y) = N C) ∂F/∂x = e^x sin y + g'(y) = N D) ∂F/∂y = e^x cos y + g(y) = N

18. Solve the linear differential equation dy/dx + y = e^x.

A) y = (1/2)e^x + Ce^(-x) B) y = e^x + Ce^(-x) C) y = (1/2)e^x + Ce^x D) y = e^x + Cx

19. What is the degree of homogeneity of the function f(x, y) = x^3 + y^3?

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

20. If dy/dx = f(x,y) is homogeneous, what is the form of f(tx, ty)?

A) f(x,y) B) tf(x,y) C) t^2f(x,y) D) f(x,y)/t

21. Which of the following is NOT a method for solving first-order differential equations?

A) Separation of Variables B) Homogeneous Equations C) Exact Equations D) Laplace Transform

22. Consider the equation (x^2 - y^2)dx + 2xy dy = 0. Is this equation exact?

A) No, because ∂M/∂y = -2y and ∂N/∂x = 2y B) Yes, because ∂M/∂y = -2y and ∂N/∂x = -2y C) No, because ∂M/∂y = 2y and ∂N/∂x = 2y D) Yes, because ∂M/∂y = 2x and ∂N/∂x = 2y

23. If a differential equation Mdx + Ndy = 0 is not exact, but M and N are homogeneous functions of the same degree, what is the integrating factor?

A) 1/(Mx + Ny) B) 1/(Mx - Ny) C) 1/(Nx + My) D) 1/(Nx - My)

24. Solve the linear equation dy/dx - y = e^(2x).

A) y = (1/3)e^(2x) + Ce^x B) y = e^(2x) + Ce^x C) y = (1/3)e^(2x) + Ce^(-x) D) y = e^x + Ce^(-2x)

25. The differential equation dy/dx = f(x, y) is called homogeneous if f(tx, ty) equals what for any scalar t?

A) f(x, y) B) t * f(x, y) C) f(x, y) / t D) t^2 * f(x, y)

26. Let dy/dx = f(y/x). After substituting y=vx, which type of differential equation do we obtain in terms of v and x?

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

27. For the equation (x^2 + y^2)dx + (2xy + y^2)dy = 0, check for exactness.

A) Not exact: ∂M/∂y = 2y, ∂N/∂x = 2x B) Exact: ∂M/∂y = 2y, ∂N/∂x = 2y C) Not exact: ∂M/∂y = 2x, ∂N/∂x = 2y D) Exact: ∂M/∂y = 2x, ∂N/∂x = 2x

28. If Mdx + Ndy = 0 is exact, then the solution can be found by integrating M with respect to x and N with respect to y, then combining terms.

A) This statement describes finding the potential function, not the final solution directly. B) This statement is correct and leads directly to the solution. C) This is only true for separable equations. D) This is incorrect; only one integration is needed.

29. What is the integrating factor for the linear equation dy/dx + (2 tan x)y = sin x?

A) sec^2(x) B) tan^2(x) C) cos^2(x) D) sin^2(x)

30. Solve the homogeneous equation dy/dx = (x+y)/x.

A) y = x ln(Cx) B) y = x ln(C/x) C) y = x ln(x/C) D) y = x ln(C)

31. The function M(x, y) = x^2y is homogeneous of what degree?

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

32. Consider the differential equation (x^2 + y^2)dx - 2xy dy = 0. Is it homogeneous?

A) Yes, M and N are homogeneous of degree 2 B) No, M is degree 2, N is degree 3 C) No, M is degree 2, N is degree 1 D) It is not homogeneous

33. For the exact equation (2xy + 1)dx + (x^2 + 4y)dy = 0, we have M = 2xy + 1 and N = x^2 + 4y. Verify exactness.

A) ∂M/∂y = 2x, ∂N/∂x = 2x (Exact) B) ∂M/∂y = 2y, ∂N/∂x = 2x (Not Exact) C) ∂M/∂y = 2x, ∂N/∂x = 2y (Not Exact) D) ∂M/∂y = 2y, ∂N/∂x = 2y (Exact)

34. When solving an exact equation Mdx + Ndy = 0, if we integrate M with respect to x to get F(x, y) = ∫Mdx + g(y), what is the relation for g'(y)?

A) g'(y) = N - ∂/∂y (∫Mdx) B) g'(y) = M - ∂/∂x (∫Ndy) C) g'(y) = N + ∂/∂y (∫Mdx) D) g'(y) = ∂/∂y (∫Mdx) - N

35. Solve the linear differential equation dy/dx - 2y = 4.

A) y = -2 + Ce^(2x) B) y = 2 + Ce^(2x) C) y = -2 + Ce^(-2x) D) y = 2 + Ce^(-2x)

36. If dy/dx = f(x,y) is homogeneous, and we substitute y=vx, what is the resulting equation in terms of v and x?

A) x dv/dx + v = f(1, v) B) x dv/dx = f(1, v) - v C) v dx/dv + x = f(v, 1) D) x dv/dx + v = f(v, 1)

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

A) y = Cx B) y = Cx^2 C) y = C/x D) y = C

38. Consider the equation (x+y)dx + (x-y)dy = 0. Is it exact?

A) No, ∂M/∂y = 1, ∂N/∂x = 1 B) Yes, ∂M/∂y = 1, ∂N/∂x = 1 C) No, ∂M/∂y = 1, ∂N/∂x = -1 D) Yes, ∂M/∂y = -1, ∂N/∂x = 1

39. If Mdx + Ndy = 0 is exact, and we have found F(x, y) = ∫Mdx + g(y), what is the final solution form?

A) F(x, y) = C B) ∫Mdx + ∫Ndy = C C) ∫Mdx = C D) ∫Ndy = C

40. Solve the linear equation dy/dx + (1/x)y = 3x.

A) y = x^2 + C/x B) y = x^3 + C/x C) y = x + C/x D) y = x^2 + Cx

41. Which condition must be satisfied for M(x, y)dx + N(x, y)dy = 0 to be an exact differential equation?

A) ∂M/∂y = ∂N/∂x B) ∂M/∂x = ∂N/∂y C) ∂M/∂x + ∂N/∂y = 0 D) ∂M/∂y - ∂N/∂x = 0

42. If M and N are homogeneous functions of the same degree n, and Mdx + Ndy = 0 is not exact, what is the integrating factor?

A) 1/(Mx + Ny) B) 1/(xM + yN) C) 1/(xN + yM) D) 1/(xN - yM)

43. What is the integrating factor for dy/dx + y/x = x?

A) x B) x^2 C) 1/x D) ln(x)

44. The differential equation dy/dx = f(x,y) is homogeneous if f(kx, ky) = k^n f(x,y) for some n. What is the specific condition for first-order homogeneous equations as commonly defined?

A) n = 0, so f(kx, ky) = f(x, y) B) n = 1, so f(kx, ky) = k f(x, y) C) n = -1, so f(kx, ky) = f(x, y)/k D) n = 2, so f(kx, ky) = k^2 f(x, y)

45. Consider the exact equation (2xy)dx + (x^2 + 3y^2)dy = 0. Find the potential function F(x, y) by integrating M=2xy with respect to x.

A) x^2y + g(y) B) xy^2 + h(x) C) 2x^2y + g(y) D) x^2y^2 + h(x)

46. After finding F(x, y) = x^2y + g(y) for the exact equation (2xy)dx + (x^2 + 3y^2)dy = 0, differentiate F with respect to y and equate to N to find g'(y).

A) ∂F/∂y = x^2 + g'(y) = x^2 + 3y^2 => g'(y) = 3y^2 B) ∂F/∂y = 2xy + g'(y) = x^2 + 3y^2 => g'(y) = x^2 - 2xy + 3y^2 C) ∂F/∂y = x^2 + g(y) = x^2 + 3y^2 => g(y) = 3y^2 D) ∂F/∂x = 2xy + g'(y) = x^2 + 3y^2

47. Integrate g'(y) = 3y^2 to find g(y).

A) g(y) = y^3 B) g(y) = 3y C) g(y) = y^2 D) g(y) = 3y^3

48. The solution to the exact equation (2xy)dx + (x^2 + 3y^2)dy = 0 is:

A) x^2y + y^3 = C B) x^2y + 3y^3 = C C) xy^2 + y^3 = C D) x^2y + y^2 = C

49. Solve the linear differential equation dy/dx - (1/x)y = x.

A) y = (x^2)/2 + Cx B) y = x^2 + Cx C) y = x + Cx D) y = x^3/2 + Cx

Homogeneous, Exact and Linear Equations - Online Test

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