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LANCASTER UNIVERSITY
2010 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 y′^ + (x+1) y = x^2 that satisfies y(1) = 0. [9]
A2. Perform a substitution to reduce the equation y′^ cos x + 2y sin x + y^2 sin x cos x = 0 to linear form, and hence solve this equation. [10]
A3. Solve the equation y′′^ − 6 y′^ + 10y = e^5 x^ sin 2x. [10]
A4. Solve the following pair of equations (in which x and y are functions of t ): x′^ = − 5 x + 2y, y′^ = x − 4 y. Make a sketch of the solution paths. [11]
A5. Let y 1 and y 2 be a pair of solutions to the equation 2 x y′′^ + (3 − 2 x) y′^ + 4y = 0, for x > 0. Define the Wronskian W of y 1 and y 2 and obtain a first order linear equation for W. By solving this equation or otherwise, determine W up to a constant. [10] please turn over
SECTION B
B1. (a) Suppose that a, b and c are positive constants. Show that all the solutions of the equation ay′′^ + by′^ + cy = 0 tend to 0 as x → ∞. [12] (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 (1 − x^2 ) y′′^ − 2 x y′^ + 2y = 0. [12]
B2. (a) Find a second order linear differential equation which has x^3 and e−^3 x^ as a pair of solutions. [5] (b) Let L(y) = y′′^ + py′^ + qy, where p and q are continuous functions on a closed bounded interval I. (i) Suppose y satisfies L(y) = 0 with y(x 0 ) = y′(x 0 ) = 0 for some x 0 ∈ I. Define a function E on I by E = y^2 + (y′)^2. Show that E(x 0 ) = 0, E(x) ≥ 0 and |E′(x)| ≤ KE(x) throughout I, where K ≥ 0 is some constant. Deduce that E(x) = 0 for all x in I (both x > x 0 and x < x 0 ), and hence show that y is the constant function with value 0. [12] (ii) State and prove the uniqueness theorem for solutions of the equation L(y) = f. [6] (c) Let y satisfy the equation y′^ = x + y^2 , with y(0) = 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 x^11. [7]
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