Solution of homogeneous and linear differential equations of first order - Question Bank

1. Separate variables for x(dv/dx) = (-v^2 - 1) / (v + 1). This gives:
A) (v + 1) / (v^2 + 1) dv = -dx/x
B) (v^2 + 1) / (v + 1) dv = -dx/x
C) (v + 1) / (v^2 + 1) dx = -dv/x
D) (v + 1) / (v^2 - 1) dv = -dx/x
2. Simplify x(dv/dx) = (v - 1) / (v + 1) - v. This leads to:
A) x(dv/dx) = (v - 1 - v(v + 1)) / (v + 1) = (-v^2 - 1) / (v + 1)
B) x(dv/dx) = (v - 1 + v(v + 1)) / (v + 1) = (v^2 + 2v - 1) / (v + 1)
C) x(dv/dx) = (v - 1 - v) / (v + 1) = -1 / (v + 1)
D) x(dv/dx) = (v - 1 + v) / (v + 1) = (2v - 1) / (v + 1)
3. Consider the homogeneous equation dy/dx = (y - x) / (y + x). After dividing by x, dy/dx = ((y/x) - 1) / ((y/x) + 1). Let y = vx. The equation becomes:
A) x(dv/dx) = (v - 1) / (v + 1) - v
B) x(dv/dx) = (v - 1) / (v + 1) + v
C) x(dv/dx) = v - (v - 1) / (v + 1)
D) x(dv/dx) = (v + 1) / (v - 1) - v
4. Bernoulli's equation dy/dx + P(x)y = Q(x)y^n can be reduced to a linear equation by substituting v = y^(1-n). What is the resulting linear equation?
A) dv/dx + (1-n)P(x)v = (1-n)Q(x)
B) dv/dx + P(x)v = Q(x)
C) dv/dx + (1-n)P(x)v = Q(x)
D) dv/dx + P(x)v = (1-n)Q(x)
5. The differential equation dy/dx + P(x)y = Q(x) is linear. If we multiply by an integrating factor I(x), the equation becomes d/dx(I(x)y) = I(x)Q(x). What is I(x)?
A) e^(∫P(x)dx)
B) e^(∫Q(x)dx)
C) ∫P(x)dx
D) ∫Q(x)dx
6. The differential equation dy/dx = (x^2 + y^2) / (2xy) can be solved by letting y = vx, which transforms it into a separable equation in x and v.
A) True
B) False
C) Only if x and y are positive
D) Only if x and y are non-zero
7. Which method is NOT directly applicable for solving dy/dx + P(x)y = Q(x)?
A) Integrating factor method
B) Substitution y = vx
C) Reduction to exact equation
D) Multiplying by integrating factor
8. Which method is NOT directly applicable for solving dy/dx = f(y/x)?
A) Substitution y = vx
B) Substitution x = vy
C) Separation of variables after substitution
D) Integrating factor method
9. The solution of dy/dx + P(x)y = 0 is:
A) y = C * e^(-∫P(x)dx)
B) y = C * e^(∫P(x)dx)
C) y = ∫P(x)dx + C
D) y = C
10. For a linear differential equation dy/dx + P(x)y = Q(x), if Q(x) = 0, the equation is called:
A) Homogeneous linear
B) Non-homogeneous linear
C) Separable
D) Exact
11. Consider dy/dx = (x^2 + xy + y^2) / x^2. This is a homogeneous equation because f(tx, ty) = t^2(x^2 + xy + y^2) / t^2x^2 = f(x, y). The degree is 0.
A) True, it's homogeneous of degree 0
B) False, the degree is not 0
C) False, it's not homogeneous
D) True, but the degree is 2
12. A first-order differential equation is called linear if it is linear in:
A) The dependent variable and its derivative
B) The independent variable
C) Both independent and dependent variables
D) The coefficients
13. What is the form of equation dy/dx = f(x, y) if f(tx, ty) = f(x, y)?
A) Homogeneous
B) Linear
C) Separable
D) Exact
14. The general solution for dy/dx + y/x = x^2 is:
A) y = x^3/4 + C/x
B) y = x^4/4x + C/x
C) y = x^3/4
D) y = x^4/4
15. The equation becomes d/dx(xy) = x^3. Integrate both sides.
A) xy = x^4/4 + C
B) xy = x^3/3 + C
C) xy = x^2/2 + C
D) xy = x^5/5 + C
16. Multiply the equation dy/dx + y/x = x^2 by the integrating factor x. The left side becomes the derivative of:
A) xy
B) y/x
C) x^2y
D) y
17. Calculate the integrating factor for dy/dx + y/x = x^2.
A) x
B) x^2
C) ln|x|
D) e^x
18. In dy/dx + y/x = x^2, what is P(x) and Q(x)?
A) P(x) = 1/x, Q(x) = x^2
B) P(x) = x, Q(x) = x^2
C) P(x) = 1, Q(x) = x^2/x
D) P(x) = x^2, Q(x) = 1/x
19. What type of equation is dy/dx + y/x = x^2?
A) Homogeneous
B) Linear
C) Bernoulli
D) Separable
20. The general solution for dy/dx = (x^2 - y^2) / (2xy) is:
A) x^3 - 3xy^2 = C
B) x^3 - 3x^2y = C
C) x - 3y^2 = C
D) x^3 - 3y^2 = C
21. Substitute back v = y/x into 1 - 3v^2 = A/x^3. This gives:
A) 1 - 3(y/x)^2 = A/x^3
B) 1 - 3(y^2/x^2) = A/x^3
C) 1 - 3y^2/x^2 = A/x^3
D) x^3 - 3x y^2 = A
22. Exponentiate both sides: |1 - 3v^2| = e^(-3 ln|x| - 3C) = e^(-3C) * e^(ln|x^-3|) = A * |x^-3|.
A) 1 - 3v^2 = A/x^3
B) 1 - 3v^2 = A*x^3
C) 1 - 3v^2 = A*x^-3
D) 1 - 3v^2 = A*x^2
23. Equating the integrals, -1/3 ln|1 - 3v^2| = ln|x| + C. Rearranging gives:
A) ln|1 - 3v^2| = -3 ln|x| - 3C
B) ln|1 - 3v^2| = 3 ln|x| + 3C
C) ln|1 - 3v^2| = -1/3 ln|x| + C
D) ln|1 - 3v^2| = ln|x| - 3C
24. Integrate dx/x. The result is:
A) ln|x| + C
B) x + C
C) ln|x|
D) 1/x + C
25. Integrate (2v / (1 - 3v^2)) dv. Using u-substitution u = 1 - 3v^2, du = 6v dv. The integral is:
A) -1/3 ln|1 - 3v^2|
B) 1/3 ln|1 - 3v^2|
C) ln|1 - 3v^2|
D) -3 ln|1 - 3v^2|
26. Separate variables for x(dv/dx) = (1 - 3v^2) / (2v). This gives:
A) (2v / (1 - 3v^2)) dv = dx/x
B) (1 - 3v^2) / (2v) dv = dx/x
C) (2v / (1 - 3v^2)) dx = dv/x
D) (2v / (1 - v^2)) dv = dx/x
27. Simplify x(dv/dx) = (1 - v^2) / (2v) - v. This leads to:
A) x(dv/dx) = (1 - 3v^2) / (2v)
B) x(dv/dx) = (1 - v^2 - 2v^2) / (2v)
C) x(dv/dx) = (1 - v^2 - 2v) / (2v)
D) x(dv/dx) = (1 - v^2) / (2v^2)
28. Let dy/dx = (1 - (y/x)^2) / (2(y/x)). Substitute y=vx. What is the transformed equation in terms of x and v?
A) x(dv/dx) = (1 - v^2) / (2v) - v
B) x(dv/dx) = (1 - v^2) / (2v) + v
C) x(dv/dx) = v - (1 - v^2) / (2v)
D) x(dv/dx) = (1 - v^2) / (2v)
29. Consider the homogeneous equation dy/dx = (x^2 - y^2) / (2xy). What is the form after dividing numerator and denominator by x^2?
A) dy/dx = (1 - (y/x)^2) / (2(y/x))
B) dy/dx = (x^2 - y^2) / (2x^2y)
C) dy/dx = (1 - y^2/x^2) / (2y/x)
D) dy/dx = (1 - (y/x)^2) / (2x/y)
30. Substitute back y = 1/v to find the general solution of dy/dx - y = xy^2.
A) 1/y = -x + 1 + Ce^-x
B) 1/y = x - 1 + Ce^-x
C) 1/y = -x - 1 + Ce^-x
D) 1/y = x + 1 + Ce^-x
31. Solve the linear equation dv/dx + v = -x. What is the general solution for v?
A) v = -x + 1 + Ce^-x
B) v = x - 1 + Ce^-x
C) v = -x - 1 + Ce^-x
D) v = x + 1 + Ce^-x
32. Solve the linear equation dv/dx + v = -x. What is the integrating factor?
A) e^x
B) e^-x
C) x
D) e^x / x
33. After substitution v = 1/y, the Bernoulli equation dy/dx - y = xy^2 becomes:
A) dv/dx + v = -x
B) dv/dx - v = x
C) dv/dx + v = x
D) dv/dx - v = -x
34. Solve the Bernoulli equation dy/dx - y = xy^2. What is the substitution to make it linear?
A) v = 1/y
B) v = y
C) v = y^-1
D) v = y^-2
35. Solve the linear differential equation dy/dx + (2/x)y = x. What is the general solution?
A) y = x^3/4 + C/x^2
B) y = x^2/3 + C/x
C) y = x^4/5 + C/x^2
D) y = x^3/5 + C/x^2
36. Solve the linear differential equation dy/dx + (2/x)y = x. What is the integrating factor?
A) x^2
B) x
C) 2x
D) x^3
37. What substitution is used to convert Bernoulli's equation into a linear differential equation?
A) v = y^(1-n)
B) v = y^n
C) v = 1/y
D) v = y
38. What is the standard form of Bernoulli's differential equation?
A) dy/dx + P(x)y = Q(x)y^n
B) dy/dx = f(y/x)
C) Mdx + Ndy = 0
D) dy/dx + P(x) = Q(x)y
39. Which type of differential equation can be reduced to a linear form by a suitable substitution?
A) Homogeneous differential equation
B) Exact differential equation
C) Bernoulli's differential equation
D) Separable differential equation
40. What is the solution of the linear differential equation dy/dx + P(x)y = Q(x) after multiplying by the integrating factor?
A) y * I.F. = ∫(Q(x) * I.F.)dx + C
B) y = ∫(Q(x) * I.F.)dx + C
C) I.F. = ∫(Q(x) * y)dx + C
D) y = ∫Q(x)dx + C
41. What is the integrating factor for the linear differential equation dy/dx + P(x)y = Q(x)?
A) e^(∫P(x)dx)
B) ∫P(x)dx
C) e^(∫Q(x)dx)
D) ∫Q(x)dx
42. What is the general form of a first-order linear differential equation?
A) dy/dx = f(x, y)
B) dy/dx = f(y/x)
C) dy/dx + P(x)y = Q(x)
D) M(x, y)dx + N(x, y)dy = 0
43. For the differential equation dy/dx = f(y/x), after substituting y=vx and differentiating y=vx with respect to x, we get dy/dx = v + x(dv/dx). This is a key step for:
A) Solving linear differential equations
B) Solving homogeneous differential equations
C) Solving exact differential equations
D) Solving separable differential equations
44. Solve the homogeneous differential equation dy/dx = (x+y)/x. What is the general solution?
A) y = x ln|x| + cx
B) y = x^2 ln|x| + cx
C) y = ln|x| + cx
D) y = cx ln|x|
45. Solve the homogeneous differential equation dy/dx = y/x. What is the general solution?
A) y = cx
B) y = cx^2
C) y = c/x
D) y = c
46. What is the condition for a differential equation M(x, y)dx + N(x, y)dy = 0 to be homogeneous?
A) M and N are homogeneous functions of the same degree.
B) M is a homogeneous function of degree n and N is a homogeneous function of degree n+1.
C) M is a homogeneous function of degree n+1 and N is a homogeneous function of degree n.
D) M and N are linear functions.
47. After substituting y = vx and rearranging, a homogeneous differential equation usually transforms into a differential equation in which variable(s)?
A) Only x
B) Only v
C) x and v
D) y and x
48. Consider the differential equation dy/dx = (x^2 + y^2) / (xy). What is the first step to solve this as a homogeneous equation?
A) Substitute y = vx
B) Substitute x = vy
C) Rewrite as dy/dx = (1 + (y/x)^2) / (y/x)
D) Separate variables
49. If we substitute y = vx into a homogeneous differential equation, what does dv/dx represent in terms of x and v?
A) The derivative of v with respect to x
B) The derivative of y with respect to x
C) The derivative of y with respect to v
D) The derivative of x with respect to v
50. Which substitution is typically used to solve a homogeneous differential equation of the form dy/dx = f(y/x)?
A) y = vx
B) x = vy
C) y = vx + c
D) x = vy + c
All Question Banks
E-Book Solution of homogeneous and linear differential equations of first order One Line Questions Solution of homogeneous and linear differential equations of first order Online Test Solution of homogeneous and linear differential equations of first order