Homogeneous, Exact and Linear Equations - Question Bank

1. 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
2. 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
3. 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
4. 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
5. 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)
6. 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)
7. What is the integrating factor for dy/dx + y/x = x?
A) x
B) x^2
C) 1/x
D) ln(x)
8. 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)
9. 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
10. 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
11. 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
12. 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
13. 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
14. 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)
15. 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)
16. 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
17. 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)
18. 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
19. The function M(x, y) = x^2y is homogeneous of what degree?
A) 3
B) 2
C) 1
D) 0
20. 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)
21. 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)
22. 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.
23. 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
24. 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
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. 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)
27. 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)
28. 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
29. 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
30. 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
31. What is the degree of homogeneity of the function f(x, y) = x^3 + y^3?
A) 3
B) 2
C) 1
D) 0
32. 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
33. 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
34. 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)
35. 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
36. 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
37. 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
38. Calculate the integrating factor for dy/dx + (1/x)y = x^2.
A) x
B) x^2
C) 1/x
D) ln(x)
39. 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
40. 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
41. 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
42. 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
43. 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
44. 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)
45. 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)
46. 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
47. 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
48. 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
49. 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
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