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LANCASTER UNIVERSITY
2011 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(x^2 + 1) y′^ − 2 y = x^2 + 1 that satisfies y(1) = −1. [10]
A2. Perform a substitution to reduce the equation y′^ sin x cosh y + 2 cos x sinh y = x to linear form, and hence solve this equation. [10]
A3. Solve the equation y′′^ − 4 y′^ + 4y = x^2 + e^2 x^ sec^2 x. [11]
A4. Solve the following pair of equations (in which x and y are functions of t ): x′^ = 2x + 2y, y′^ = 3x + y. Make a sketch of the solution paths. [11]
A5. Let y satisfy the equation y′^ = 1+x^2 y^2 , with y(0) = 1. 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 and including a term in x^9. [8] please turn over
SECTION B
B1. (a) The function y satisfies y(1) = 1 and x y′^ − 2 y ≥ x^2 − 2 for all x > 0. Show that y(x) ≥ 1 + x^2 log x for x ≥ 1. [8] (b) Let L(y) = y′′^ + py′^ + qy, where p and q are continuous functions on an interval I. Let y 1 and y 2 be two solutions of L(y) = 0 and let W = y 1 y 2 ′ − y 2 y 1 ′ be the Wronskian of y 1 and y 2. (i) Show that W satisfies the first order equation W ′^ + pW = 0, and hence derive an expression for W in terms of P (x), where P ′(x) = p(x). Deduce that if W is zero at one point of I, then it is zero throughout I. [8] (ii) Suppose that W is not zero throughout I. Suppose also that x 1 and x 2 are points of I such that x 1 < x 2 , y 1 (x 1 ) = 0 = y 1 (x 2 ) and y 1 (x) > 0 for x 1 < x < x 2. Show that y 1 ′(x 1 ) ≥ 0 and y′ 1 (x 2 ) ≤ 0. By considering the behaviour of W on [x 1 , x 2 ], prove that y 2 must have a zero in (x 1 , x 2 ). [9] (c) Let y 1 and y 2 be a pair of solutions to the equation (1 − x^2 ) y′′^ − 2 x y′^ + 6y = 0 on the interval (− 1 , 1). Calculate the Wronskian of y 1 and y 2 up to a multiplicative constant. [5]
B2. (a) In the equation x y′′^ − 2(x + 1) y′^ + 2(x + 1) y = 0 write y = uv. Show how to choose u so that v satisfies the equation v′′^ +
1 − (^) x^22
v = 0 (∗) with no v′^ term. [7] (b) State Sturm’s Comparison Theorem. Let v be a non-trivial solution of equation (∗). Show that v has at most two zeros in the interval (0, 2 π). [7] (c) Let L(y) = − 4 y′′^ − y and consider the Sturm–Liouville system given by the equation L(y) = λy on the interval I = [0, 1], together with the boundary conditions y′(0) = 0, y(1) = 0. Find all of the eigenvalues and corresponding eigenfunctions. [16] please turn over
