THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS November, 2018 MATH2121 Theory and Applications of Differential Equations (1) TIME ALLOWED – 2 hours (2) TOTAL NUMBER OF QUESTIONS – 4 (3) ANSWER ALL QUESTIONS (4) THE QUESTIONS ARE OF EQUAL VALUE (5) ANSWER EACH QUESTION IN A SEPARATE BOOK (6) THIS PAPER MAY BE RETAINED BY THE CANDIDATE (7) ONLY CALCULATORS WITH AN AFFIXED “UNSW APPROVED” STICKER MAY BE USED All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing, sketching or graphical work. November, 2018 MATH2121 Page 2 Bessel equation The Bessel equation of order ν is x2y′′ + xy′ + (x2 − ν2)y = 0. The Bessel function of the first kind is Jν(x) = ∞∑ k=0 (−1)k k!Γ(k + ν + 1) (x 2 )2k+ν . For x→ 0+, Jν(x)→ 0, ν > 0 and J0(x)→ 1. For x ∈ R and x > 0, d dx (xνJν(x)) = x νJν−1(x). A Sturm-Liouville equation is of the form (p(x)y′)′ + (q(x) + λr(x)) y = 0. Parseval’s Identity states 1 L ∫ L −L f 2(x) dx = a20 2 + ∞∑ n=1 ( a2n + b 2 n ) . Variation of parameters y(x) = −y1(x) ∫ y2(x)f(x) W (x) dx+ y2(x) ∫ y1(x)f(x) W (x) dx. Please see over . . . November, 2018 MATH2121 Page 3 1. i) Use the variation of parameters method to solve y′′ − 2y′ + y = ex cos(x). ii) Solve the following ODEs a) x2y′ + 3xy = sin(x) x , b) x dy dx = y ( 1 + ln (y x )) . iii) We aim to construct a series solution to the ODE about the ordinary point x0 = 0 (1− x2)y′′ − 2xy′ + 20y = 0, y(0) = 1, y′(0) = 0, of the form y(x) = ∞∑ n=0 Anx n. (1) a) Give the recurrence relation for the coefficients An. b) Explain from the recurrence relation that one of the series will ter- minate yielding a polynomial solution, and the other does not. c) Write down the polynomial solution. 2. i) Given y1 = e 2x is a solution to the following ODE, find the general solu- tion xy′′ − (1 + 2x)y′ + 2y = 0. ii) Find the general solution to the following ODE x2y′′ − 3xy′ + 4y = 0. Does a solution exist such that y(0) = 1 can be satisfied? iii) Solve for x(t) and y(t) and determine the type and stability of the equi- librium point of the following system of differential equations: dx dt = x+ y ; dy dt = 2x. Please see over . . . November, 2018 MATH2121 Page 4 3. i) Consider the ODE y′′ + 2y′ + (1− λ)y = 0, y(0) = 0, y(1) = 0, λ < 0. a) Transform the equation into Sturm-Liouville form. b) Find the eigenvalues and eigenfunctions. c) State the orthogonality condition satisfied by the eigenfunctions. ii) a) Write down the general solution to x2y′′ + xy′ + (5x2 − 3)y = 0. b) Evaluate the integral ∫ x8J3(x) dx. 4. Consider a vibrating string of length 1 undergoing transverse displacement u(x, t) according to the wave equation ∂2u ∂t2 = ∂2u ∂x2 , at position x and time t. The ends of the string are held fixed so that u(0, t) = u(1, t) = 0, ∀t. Using the method of separation of variables, let u(x, t) = F (x)G(t), and i) derive the following differential equations, G′′ − kG = 0, F ′′ − kF = 0, where k is the separation constant. ii) You may assume that only k = −w2 < 0 yields non-trivial solutions. Apply the boundary conditions and solve for F (x). iii) Find all possible solutions Gn(t) for G(t). iv) The initial displacement and velocity of the string are u(x, 0) = f(x), ∂u ∂t (x, 0) = 0, where f(x) = { 2x, 0 ≤ x ≤ 1/2, 2(1− x), 1/2 < x ≤ 1. Write down the general solution u(x, t). END OF EXAMINATION
