Pair of Equations - Differential Equations - Exam, Exams of Differential Equations

Download Pair of Equations - Differential Equations - Exam and more Exams Differential Equations in PDF only on Docsity! LANCASTER UNIVERSITY 2009 EXAMINATIONS PART II (Third Year) MATHEMATICS & STATISTICS 2 hours Math 318: Differential Equations You should answer ALL Section A questions and TWO Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. SECTION A A1. Find the solution of the equation (x log x) y′ + y = 1 + log x that satisfies y(2) = 1. [10] A2. Perform a substitution to reduce the equation y′ cos x cos y − 2 sin x sin y = cos x to linear form, and hence solve this equation. [11] A3. Solve the equation x2 y′′ − 5x y′ + 8y = x + x3. [9] A4. Solve the following pair of equations (in which x and y are functions of t ): x′ = 5x + 3y, y′ = x + 7y. Make a sketch of the solution paths. [12] A5. Let y satisfy the equation y′ = 2x − 1 − y3, with y(1) = 0. Write down the iterative formula used in Picard’s scheme for finding approximations to y. Use this scheme to find an approximation to y up to a term in x7. [8] please turn over 1 SECTION B B1. (a) Suppose that y satisfies y′′−2my′ +m2y = 0 for some constant m, and let z = y′−my. Derive a first order equation satisfied by z. Using integrating factors, find the general expression for y. [8] (b) Let L(y) = py′′ + qy′ + ry, where p, q and r are functions. Suppose that u satisfies L(u) = 0 and y = uv is required to satisfy L(y) = f . Derive the equation satisfied by v. Also state the condition for u = x to satisfy L(u) = 0. [6] Solve the equation x2 y′′ − 2x(x+1) y′ + 2(x+1) y = x4. [11] (c) Find a second order linear differential equation which has xex and e2x as a pair of solutions. [5] B2. (a) The function y satisfies y(0) = 0 and (x+1) y′ − 2y ≥ x for all x > −1. Show that y(x) ≥ x2/2 for x ≥ 0. [8] (b) Suppose that functions y and z satisfy y′′ + q(x) y = 0, z′′ + r(x) z = 0 on an interval I, where q and r are functions such that q(x) > r(x) for x in I. Suppose that x1, x2 are points of I such that z(x1) = z(x2) = 0 and z(x) > 0 for x1 < x < x2. Show that z′(x1) ≥ 0 and z′(x2) ≤ 0. Let W = yz′ − y′z. By considering the behaviour of W on [x1, x2], prove that y must have a zero in (x1, x2). [16] Let z be a non-trivial solution of 4xz′′ + z = 0. By comparison with x1/2 (or otherwise) show that z has at most one zero in (0, 1). [6] please turn over 2