TRB PG Assistant - Total differential equations, first-order partial differential equations, Charpit's method Online Test (MCQ Quiz)

1. What is the condition for a differential equation M dx + N dy = 0 to be a total differential equation?

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

2. If M dx + N dy = 0 is a total differential equation, what is its general solution?

A) ∫M dx + ∫N dy = C B) ∫M dx + ∫(N - ∫(∂M/∂y) dx) dy = C C) ∫M dx + ∫N dy - ∫(∂M/∂y) dx dy = C D) ∫M dx + ∫(N - ∂M/∂y) dy = C

3. Consider the equation (x² + y) dx + (x - y) dy = 0. Is this a total differential equation?

A) Yes, because ∂M/∂y = 1 and ∂N/∂x = 1 B) No, because ∂M/∂y = 1 and ∂N/∂x = 1 C) Yes, because ∂M/∂x = 2x and ∂N/∂y = -1 D) No, because ∂M/∂x = 2x and ∂N/∂y = -1

4. Solve the total differential equation (2xy + 1) dx + (x² - 2y) dy = 0.

A) x²y + x - y² = C B) x²y + y - x² = C C) xy² + x - y² = C D) x²y + x - y = C

5. What is the integrating factor for the equation y dx - x dy = 0?

A) 1/y² B) 1/x² C) 1/(x² + y²) D) 1/(xy)

6. If M dx + N dy = 0 is not exact, and (∂M/∂y - ∂N/∂x)/(N) is a function of x only, say f(x), what is the integrating factor?

A) e^∫f(x) dx B) e^-∫f(x) dx C) ∫f(x) dx D) 1/f(x)

7. If M dx + N dy = 0 is not exact, and (∂N/∂x - ∂M/∂y)/(M) is a function of y only, say g(y), what is the integrating factor?

A) e^∫g(y) dy B) e^-∫g(y) dy C) ∫g(y) dy D) 1/g(y)

8. What is the order of a partial differential equation?

A) The highest order of derivative present in the equation B) The number of dependent variables C) The number of independent variables D) The degree of the highest order derivative

9. What is the degree of a partial differential equation?

A) The highest order of derivative present in the equation B) The number of dependent variables C) The number of independent variables D) The highest power of the highest order derivative

10. Which of the following is a first-order partial differential equation?

A) ∂²u/∂x² + ∂²u/∂y² = 0 B) ∂u/∂x + ∂u/∂y = u C) ∂³u/∂x³ = sin(x) D) u = x ∂u/∂x + y ∂u/∂y

11. What is a quasi-linear first-order partial differential equation of the form P(x, y, u) ∂u/∂x + Q(x, y, u) ∂u/∂y = R(x, y, u)?

A) P, Q, and R are functions of x, y, and u. B) ∂u/∂x and ∂u/∂y are linear, and P, Q, R can be any functions. C) P and Q are functions of x and y only, and R is a function of u only. D) P, Q, and R are functions of x and y only.

12. What are the characteristic equations for the first-order PDE P(x, y, u) ∂u/∂x + Q(x, y, u) ∂u/∂y = R(x, y, u)?

A) dx/P = dy/Q = du/R B) dx/R = dy/Q = du/P C) dx/Q = dy/P = du/R D) dx/P = dy/R = du/Q

13. If the characteristic equations are dx/P = dy/Q = du/R, and we find two independent integrals f(x, y, u) = c1 and g(x, y, u) = c2, what is the general solution of the PDE?

A) f(x, y, u) = c1 B) g(x, y, u) = c2 C) Φ(f, g) = 0, where Φ is an arbitrary function D) f + g = C

14. Solve the PDE ∂u/∂x + 2 ∂u/∂y = u using the method of characteristics.

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

15. What is Charpit's method used for?

A) Solving linear first-order PDEs B) Solving quasi-linear first-order PDEs C) Solving non-linear first-order PDEs D) Solving second-order PDEs

16. Charpit's auxiliary equations for a non-linear first-order PDE F(x, y, u, p, q) = 0, where p = ∂u/∂x and q = ∂u/∂y, are given by:

A) dx/F_p = dy/F_q = du/(p F_p + q F_q) = dp/(F_x + p F_u) = dq/(F_y + q F_u) B) dx/F_x = dy/F_y = du/(F_u) = dp/(F_p) = dq/(F_q) C) dx/p = dy/q = du/(F_x + F_y) D) dx/F_u = dy/F_u = du/(p F_p + q F_q)

17. In Charpit's method, we aim to find a relation between p and q such that F(x, y, u, p, q) = 0 can be reduced to the form:

A) ∂u/∂x = f(x, y) B) ∂u/∂y = g(x, y) C) ∂u/∂x = f(x, a) or ∂u/∂y = g(y, a), where 'a' is a constant D) ∂u/∂x + ∂u/∂y = h(x, y)

18. Consider the non-linear PDE p² + q² = 1. What are Charpit's auxiliary equations?

A) dx/2p = dy/2q = du/(2p² + 2q²) = dp/0 = dq/0 B) dx/p = dy/q = du/(p² + q²) = dp/0 = dq/0 C) dx/0 = dy/0 = du/(p² + q²) = dp/2p = dq/2q D) dx/1 = dy/1 = du/(p+q) = dp/0 = dq/0

19. Using Charpit's method for p² + q² = 1, we find dp = 0 and dq = 0. This implies p = a and q = b, where a² + b² = 1. What is the form of the solution?

A) u = ax + by + C B) u = ax + √(1-a²)y + C C) u = √ax + √by + C D) u = ax + y + C

20. Solve the non-linear PDE q = xp + p². What is a particular integral using Charpit's method?

A) u = ax + a²y + b B) u = ax + a²x + b C) u = ax + y + a² D) u = ax + a²y

21. Which of the following is a first-order partial differential equation?

A) y''(x) + y(x) = 0 B) ∂u/∂x + ∂u/∂y = 0 C) ∂²u/∂x² - ∂²u/∂y² = 0 D) ∇²u = 0

22. The equation ∂u/∂x + ∂u/∂y = 0 is:

A) A second-order linear PDE B) A first-order linear homogeneous PDE C) A first-order non-linear PDE D) A first-order quasi-linear PDE

23. What is the characteristic equation for ∂u/∂x + ∂u/∂y = 0?

A) dx/1 = dy/1 = du/0 B) dx/0 = dy/1 = du/1 C) dx/1 = dy/0 = du/1 D) dx/1 = dy/1 = du/1

24. From the characteristic equations dx/1 = dy/1, we get dy = dx. Integrating this gives:

A) y = x + c B) y = -x + c C) x + y = c D) x - y = c

25. From the characteristic equations dy/1 = du/0, we get du = 0. Integrating this gives:

A) u = c B) u = 0 C) u = x D) u = y

26. The general solution of ∂u/∂x + ∂u/∂y = 0 is of the form:

A) u = Φ(x + y) B) u = Φ(x - y) C) u = Φ(y - x) D) u = Φ(x) + Φ(y)

27. Consider the PDE x ∂u/∂x + y ∂u/∂y = u. This is a:

A) First-order linear homogeneous PDE B) First-order linear non-homogeneous PDE C) First-order non-linear PDE D) Second-order PDE

28. What are the characteristic equations for x ∂u/∂x + y ∂u/∂y = u?

A) dx/x = dy/y = du/u B) dx/y = dy/x = du/u C) dx/u = dy/u = du/(x+y) D) dx/x = dy/y = du/0

29. From dx/x = dy/y, we get the integral:

A) x/y = c1 B) y/x = c1 C) xy = c1 D) x + y = c1

30. From dy/y = du/u, we get the integral:

A) y/u = c2 B) u/y = c2 C) yu = c2 D) y + u = c2

31. The general solution of x ∂u/∂x + y ∂u/∂y = u is of the form:

A) u = y Φ(x/y) B) u = x Φ(y/x) C) u = Φ(x/y) D) u = Φ(y/x)

32. What is the Lagrange-Charpit's method for solving first-order PDEs?

A) A method for solving linear PDEs of any order. B) A method for solving quasi-linear PDEs. C) A general method for solving non-linear first-order PDEs. D) A method for solving second-order PDEs.

33. Charpit's method requires finding an integral of the form:

A) ∂u/∂x = f(x, y, u) B) ∂u/∂y = g(x, y, u) C) ∂u/∂x = f(x, a) or ∂u/∂y = g(y, a) D) ∂u/∂x + ∂u/∂y = h(x, y, u)

34. Consider the PDE pq = x + y. Using Charpit's method, if we choose dp/dq = q/p, what is the relation between p and q?

A) p² = q² + C B) p = q² + C C) p² = q + C D) p = q + C

35. For the PDE pq = x + y, if we choose p² = q² + C, and let C = 0 (so p=q), what is the relation between p and q?

A) p = q = √(x+y) B) p = q = x+y C) p = q = √(x) + √(y) D) p = q = x

36. If p = q = √(x+y) for the PDE pq = x + y, what is the form of the differential equation to solve for u?

A) ∂u/∂x + ∂u/∂y = √(x+y) B) ∂u/∂x = √(x+y) C) ∂u/∂y = √(x+y) D) ∂u/∂x + ∂u/∂y = x+y

37. Integrating ∂u/∂x = √(x+y) with respect to x, treating y as a constant, yields:

A) u = (2/3)(x+y)^(3/2) + Φ(y) B) u = (3/2)(x+y)^(2/3) + Φ(y) C) u = √(x+y) + Φ(y) D) u = x√(x+y) + Φ(y)

38. If M dx + N dy + P dz = 0 is a total differential equation, what is the condition for integrability?

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

39. What is the geometrical interpretation of the method of characteristics for first-order PDEs?

A) Finding curves along which the PDE becomes an ODE. B) Finding surfaces that satisfy the PDE. C) Finding singularities of the PDE. D) Finding stationary points of the solution.

40. The equation z = f(x, y) is a solution to a first-order PDE if:

A) It satisfies the PDE for some arbitrary function f. B) It satisfies the PDE for any arbitrary function f. C) It satisfies the PDE for some constants. D) It satisfies the PDE for some specific values of x and y.

41. Consider the PDE ∂u/∂x = 0. What is its general solution?

A) u(x, y) = C B) u(x, y) = Φ(y) C) u(x, y) = Φ(x) D) u(x, y) = x + Φ(y)

42. Consider the PDE ∂u/∂y = 0. What is its general solution?

A) u(x, y) = C B) u(x, y) = Φ(y) C) u(x, y) = Φ(x) D) u(x, y) = y + Φ(x)

43. The equation ∂u/∂x + ∂u/∂y = 0 has characteristics that are:

A) Parallel to the x-axis B) Parallel to the y-axis C) Lines with slope 1 (y = x + c) D) Lines with slope -1 (y = -x + c)

44. The equation x ∂u/∂x + y ∂u/∂y = 0 has characteristics that are:

A) Lines through the origin (y = mx) B) Circles centered at the origin (x² + y² = r²) C) Hyperbolas (xy = c) D) Parabolas (y = ax²)

45. Charpit's method is particularly useful when the PDE F(x, y, u, p, q) = 0 is:

A) Linear and homogeneous B) Linear and non-homogeneous C) Non-linear, but can be simplified by finding a relation between p and q D) Of second order

46. If we have found two independent integrals f(x, y, u) = c1 and g(x, y, u) = c2 from the characteristic equations of a first-order PDE, the general solution is given by:

A) f = g B) f + g = C C) Φ(f, g) = 0 D) f * g = C

47. Consider the total differential equation (y² + xy) dx + (xy + x²) dy = 0. Is it exact?

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

48. Solve the exact differential equation (y² + xy) dx + (xy + x²) dy = 0.

A) x²y + xy² = C B) xy² + x²y = C C) x²y² + xy = C D) xy(x+y) = C

Total differential equations, first-order partial differential equations, Charpit's method - Online Test

Test Results