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LANCASTER UNIVERSITY
2008 EXAMINATIONS
PART II (Third or Fourth 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
y′^ cosh x − 2 y sinh x = sinh x cosh x
that satisfies y(0) = 2. [10]
A2. Perform a substitution to reduce the equation
(x^2 + 1) y′^ = x y − 2 x y^4
to linear form, and hence solve this equation. [11]
A3. Solve the equation y′′^ + 3y′^ − 4 y = 5x ex. [9]
A4. Solve the following pair of equations (in which x and y are functions of t ):
x′^ = 4x + 3y, y′^ = 2x − y.
Make a sketch of the solution paths. [12]
A5. The function y satisfies y(0) = 0 and y′^ ≥ 4 x(y + 1) for all x. Prove that y ≥ e^2 x^2 − 1 for all x ≥ 0. [8]
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SECTION B
B1. (a) 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. [5] Solve the equation x^2 y′′^ − 3 x y′^ + 4y = log x. [12]
(b) Find a second order linear differential equation which has x + 1 and e−x^2 as a pair of solutions. [5] (c) State the uniqueness theorem for solutions of a second-order linear differential equation. Explain why there do not exist continuous functions p and q defined on R such that the equation y′′^ + py′^ + qy = 0 has x + 1 and ex^ as a pair of solutions. [8]
B2. (a) Obtain two linearly independent series solutions of the equation
y′′^ + 4xy = 0. (*)
Calculate the first four non-zero terms of each series explicitly. Also determine the radius of convergence of each series. [16] (b) (i) State Sturm’s Comparison Theorem. [3] Let y be a non-trivial solution of equation (*). (ii) Show that y has at most one zero in the interval (−∞, 0). [5] (ii) Let n be a positive integer and suppose x 0 > 14 n^2. Show that y has at least n zeros in the interval (x 0 , x 0 + π). [6]
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