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LANCASTER UNIVERSITY
2007 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
(x^2 + 1) y′^ = 2x y + x^3
that satisfies y(0) = 1. [11]
A2. Perform a substitution to reduce the equation
x y′^ + 6y = 3x y^4 /^3
to linear form, and hence solve this equation. [8]
A3. Solve the equation y′′^ + 2y′^ − 3 y = (5x^2 + 2x) e^2 x. [11]
A4. Solve the following pair of equations (in which x and y are functions of t ):
x′^ = 6x + 3y, y′^ = 4x + 5y.
Make a sketch of the solution paths. [12]
A5. Let y satisfy the equation y′^ = x^2 + y^2 , with y(0) = 1. Use Picard’s iteration to find an approximation to y, up to a term in x^7. [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 y′′^ + y = tan x. [12]
(b) In the equation x y′′^ + y′^ + (4x + 1) y = 0 write y = uv. Show how to choose u so that v satisfies the equation
v′′^ +
4 + 1 x + 41 x 2
v = 0 (∗)
[6] (c) State Sturm’s Comparison Theorem. Let v be a non-trivial solution of equation (∗). Show that v has at least two zeros in the interval (π, 2 π). [7]
B2. (a) Obtain two linearly independent series solutions of the equation
y′′^ − 3 x^2 y′^ − 6 xy = 0
Calculate the first four non-zero terms of each series explicitly. Also determine the radius of convergence of each series and express one of the series in terms of elementary functions. [18] (b) Find two values of ρ for which the equation
2 x y′′^ + (1 + x) y′^ − 2 y = 0
has a solution of the form y = xρ
∑^ ∞
n=
anxn, with a 0 = 1. In each case express an in terms of an− 1. Find one polynomial solution. (You do not need to write out the other series explicitly.) [12]
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