Exam 10 January 2012, questions PDF

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MATH 265001MATH 265001This question paper consists of 3 printed pages, each of which is identified by the reference MATH 265001All calculators must carry an approval sticker issued by the School of Mathematics. ©c UNIVERSITY OF LEEDS Examination for the Module MATH 2650 (January 2012)Calculus of Var...

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MATH 265001

MATH 265001 All calculators must carry an approval sticker issued by the School of Mathematics.

This question paper consists of 3 printed pages, each of which is identified by the reference MATH 265001

c UNIVERSITY OF LEEDS  Examination for the Module MATH 2650 (January 2012) Calculus of Variations Time allowed: 2 hours Answer no more than four questions. All questions carry equal marks.

1. (a) State and prove the Fundamental Lemma of the Calculus of Variations. (b) The Euler-Lagrange equation for the integral Z x2 I= F (x, y(x), y ′ (x)) dx x1

takes the form

∂F d − ∂y dx



∂F ∂y ′



= 0.

Write down a first integral of the Euler-Lagrange equation when F = F (x, y ′ ). Derive the first integral (the Beltrami identity) y′

∂F −F = C ∂y ′

(where C is an arbitrary constant) of the Euler-Lagrange equation in the case when F = F (y, y ′ ). (c) Find the functions y(x) that are extrema of the following integrals, subject to the given end conditions: Z π/2   2 (i) I = y 2 − y ′ − 2y cos x dx, y(0) = 0, y(π/2) = 1; 0 Z π/3   2 (ii) I = y ′ − y ′ tan x dx, y(0) = 0, y(π/3) = 1; Z0 2   2 (iii) I = y ′ y dx, y(0) = 0, y (2) = 4. 0

MATH 265001

2. (a) Consider the integral I=

Z

x2

F (x, y(x), y ′ (x)) dx. x1

Derive the necessary condition for I to be stationary for the following two cases: (i) When the values of y(x) at x = x1 and x = x2 are specified; (ii) When the values of y(x) at x = x1 and x = x2 are not specified. [You may assume without proof, but should state, the Fundamental Lemma of the Calculus of Variations.] (b) Find the functions y(x) that make the integral I=

Z

π/2 0



 y ′2 − y 2 − 2xy dx

stationary, for the following two cases: (i) y(0) = 0, y(π/2) = 1; (ii) No condition on y(0) or y(π/2). Show that if y(0) = 0, it is not possible to find an extremal y(x) with no condition specified for y(π/2).

3. (a) Consider the integral I=

Z

x2

F (x, y(x), y ′ (x), y ′′ (x), . . . , y (n) (x)) dx,

x1

where y (n) denotes dn y/dxn , and where y(x1 ) = A0 , y ′ (x1 ) = A1 , . . . , y (n−1)(x1 ) = An−1 ; y(x2 ) = B0 , y ′ (x2 ) = B1 , . . . , y (n−1)(x2 ) = Bn−1 . Derive the necessary condition (an extended version of the Euler-Lagrange equation) for I to be minimised. [You may assume without proof, but should state, the Fundamental Lemma of the Calculus of Variations.] (b) Find the function y(x) that minimise the integral I=

Z

π/2 0



 y ′′2 − 2y ′2 + y 2 dx,

with y(0) = 0, y(π/2) = 0, y ′ (0) = 1, y ′ (π/2) = 2.

MATH 265001

4. (a) State Fermat’s principle for the propagation of light rays. (b) Suppose c(y) is the (variable) speed of light in a medium. Show, by using Fermat’s principle and the Euler-Lagrange equation, that light rays are described by x + x0 = ±K

Z

√

c(y)dy , 1 − K 2 c2

where K and x0 are constants. (c) Find the shape of the light rays y(x) when the speed of light is given by c(y) =

a , 1 + βy

where a and β are constants. Show that the ray that passes through (0, y0 ) with y ′ (0) = 0 is given by     βx 1 (1 + βy0 ) cosh −1 . y= β 1 + βy0

5. Consider the Sturm-Liouville system composed of the ordinary differential equation   dy d + q(x)y + λr(x)y = 0, p(x) dx dx

(∗)

with boundary conditions y(x1 ) = y1 , y(x2 ) = y2 . Define Λ by Λ=

R x2 x1

(†)

(p(x)y ′2 − q(x)y 2 ) dx R x2 . r(x)y 2 dx x1

(a) Show that the problem of determining functions satisfying (†) that make Λ stationary is equivalent to the problem of determining the eigenfunctions of (∗), subject to (†). (b) Now consider the case when p(x) = x, q(x) = 1 and r(x) = 1, with boundary conditions y(0) = y(1) = 0. By using the result in part (a) and adopting the trial function y = cx(1−x), obtain an estimate for the lowest eigenvalue λ. (c) Now obtain a different estimate for the problem in part (b) by adopting the trial function y = sin πx. (d) Which of these two estimates is closest to the true eigenvalue?

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