JEE Main - Solution of homogeneous and linear differential equations of first order Online Test (MCQ Quiz)

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

A) dy/dx = f(x, y) B) dy/dx = f(ax + by + c) C) dy/dx = f(y/x) D) dy/dx + P(x)y = Q(x)

2. 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

3. 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

4. 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

5. 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

6. 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.

7. 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

8. 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|

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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)

24. 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)

25. 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)

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. 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|

28. Integrate dx/x. The result is:

A) ln|x| + C B) x + C C) ln|x| D) 1/x + C

29. 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

30. 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

31. 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

32. 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

33. What type of equation is dy/dx + y/x = x^2?

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

34. 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

35. Calculate the integrating factor for dy/dx + y/x = x^2.

A) x B) x^2 C) ln|x| D) e^x

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

45. 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

46. 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

47. 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

48. 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)

49. 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

50. 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)

Solution of homogeneous and linear differential equations of first order - Online Test

Test Results