Total differential equations, first-order partial differential equations, Charpit's method - Question Bank

1. 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
2. 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
3. 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
4. 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
5. 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)
6. 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)
7. 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)
8. 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)
9. 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.
10. 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.
11. 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
12. 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)
13. 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
14. 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
15. 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
16. 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)
17. 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.
18. 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)
19. From dy/y = du/u, we get the integral:
A) y/u = c2
B) u/y = c2
C) yu = c2
D) y + u = c2
20. From dx/x = dy/y, we get the integral:
A) x/y = c1
B) y/x = c1
C) xy = c1
D) x + y = c1
21. 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
22. 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
23. 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)
24. 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
25. 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
26. 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
27. 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
28. 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
29. 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
30. 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
31. 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
32. 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)
33. 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)
34. 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
35. 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)
36. 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
37. 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
38. 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.
39. 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
40. 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
41. 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
42. 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)
43. 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)
44. 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)
45. 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
46. 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
47. 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
48. 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
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