MTH106 Final Review Exercises
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MTH106 Final Review Exercises
Question 1-Question 8: Multiple Choice Questions (only one correct answer)
Problem 1. Consider the statements:
(i) + ex y + e_x y2 = sin x is a 1st order nonlinear ordinary differential equation. (ii) , is a system of 1st order linear ordinary differential equations.
(iii) (1 + uy(2))uxx ← 2ux uy uxy + (1 + ux(2))uyy = 0 is a 2nd order nonlinear partial differential equation. (iv) uxx + uyy = 1 ← x2 ← y2 is a 2nd order linear partial differential equation.
Which of them are true?
(A) All of them.
(B) (i), (iii) and (iv) only.
(C) (i) and (iii) only.
(D) (iv) only.
(E) None of the answers (A), (B), (C) and (D) are correct.
Problem 2. Consider the following initial value problem:
dy y 」
dt t
Compute y(2):
(A) ←2 ln(1 + ln 2).
(B) ← ln(1 + ln 2).
(C) ←2 ln(1 ← ln 2).
(D) ← ln(1 ← ln 2).
(E) None of the answers (A), (B), (C) and (D) are correct.
Problem 3. Let a, b, c, γ è R. Solve the following differential equation:
(axy2 + bx y) + (cx + γy)x22 = 0.
Consider the statements:
(i) If a = c = γ = 1 and b = 3, then the differential equation is exact and the general solution is ←x3 y = k with k è R.
(ii) If a = b = c = γ = 1, then the differential equation is exact and the general solution is + x3 y = k
with k è R.
(iii) If a = γ = 1 and b = c = 0, then the differential equation is exact and the general solution is x2 y2 = k
with k ≤ 0.
(iv) If a = γ = 1 and b = c = 0, then the differential equation is separable.
Which of them are true?
(A) All of them.
(B) (i), (iii) and (iv) only.
(C) (i) and (iii) only.
(D) (iv) only.
(E) None of the answers (A), (B), (C) and (D) are correct.
Problem 4. Which one of the following is a fundamental matrix for the system:
x-\ = 、x-.
(A) Φ(t) = 、
(B) Φ(t) = 、
(C) Φ(t) = 、
(D) Φ(t) = 、
(E) Φ(t) = ╱ ←e_(2e)8t(_) 、
Problem 5. Which one of the following is the Fourier series expansion of
f (x) = sin3 x, 0 = x = 2π
(A) sin x + sin 3x
(B) sin x ← sin 3x
(C) sin x + sin 3x
(D) sin x ← sin 3x
(E) None of the above.
Problem 6. Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are real. (Let n represent an arbitrary positive number.)
y\\ + λy = 0, y(0) = 0, y\ (π) = 0
(A) λn = and yn = sin / ←
(B) λn = and yn = sin / ←
(C) λn = and yn = sin / ←
(D) λn = and yn = sin / ←
(E) λn = and yn = sin / ←
Problem 7. Determine whether the method of separation of variables can be used to replace the given partial differential equation by a pair of ordinary differential equations.
xuxx + ut = 0
(A) Yes, the method of separation of variables can be used, and the equations are xX\\ ← λX = 0 and
T\ + λT = 0, where λ is some constant.
(B) Yes, the method of separation of variables can be used, and the equations are xX\\ ← λX = 0 and
T\ ← λT = 0, where λ is some constant.
(C) Yes, the method of separation of variables can be used, and the equations are xX\ ← λX = 0 and T\ + λT = 0, where λ is some constant.
(D) Yes, the method of separation of variables can be used, and the equations are xX\ ← λX = 0 and T\ ← λT = 0, where λ is some constant.
(E) No, the method of separation of variables cannot be used.
Problem 8. Solve u = u(x, y) for the boundary value problem:
∂u ∂u
∂x ∂y
u(0, y) = 8e_3y + 4e_5y
Which of the following is the correct answer?
(A) u(x, y) = 8e_3(4x+y) + 4e_5(4x+y);
(B) u(x, y) = 8e_4x _3y + 4e_4x _5y;
(C) 12
(D) 0
(E) None of the above.
Question 9-Question 12: Computation and Proof Questions
Problem 9. Solve the following initial value problem
dy 2
dx
by using the substitution y(x) = x + u(x).
Problem 10. Consider the linear non-homogeneous system:
x-\ = ┐ x- + ┌ 3(1) ┐ , x- = ╱ x(x)2(1) 、, t > 0.
(A) Solve the associated homogeneous linear system to find the complimentary function x-c ; (B) Solve the non-homogeneous linear system.
Problem 11. Consider the following function
f (x) =
if 0 = x < 1
if 1 = x = 2.
(i) Draw the even extension fe of f over [←2, 2].
(ii) Expand the function fe in a Fourier series.
(iii) Draw the odd extension fo of f over [←2, 2]. (Hint: you need to set fo (0) = 0.)
(iv) Expand the function fo in a Fourier series.
Problem 12. If an elastic string is free at one end, the boundary condition to be satisfied there is that ux = 0. Find the displacement u(x, t) in an elastic string of length L, fixed at x = 0 and free at x = L, set in motion with no initial velocity from the initial position u(x, 0) = f (x), where f is a given function.
(A) Formulate the mathematical model with the partial differential equation a2 uxx = utt and equip the
equation with suitable boundary conditions or/and initial conditions.
2
(B) Show that u(x, t) = An sin(λn x) cos(λn at), where λn = (2n ← 1)π/2L, n = 1, 2, . . .. Please provide
n=1
all details of your work.
(C) Show that ísin(λn x)〕is an orthogonal set on the interval [0, L], find an expression of An in (B) and hence solve the displacement un for the boundary value problem proposed in (A).
2022-08-08