Exam May, questions PDF

Preview

Summary

MATH237501 This question paper consists of 4 printed pages, each of which is identified by the reference MATH237501. All calculators must carry an approval sticker issued by the School of Mathematics. c University of Leeds School of Mathematics May/June 2017 MATH237501 Linear Differential Equations ...

Description

MATH237501

This question paper consists of 4 printed pages, each of which is identified by the reference MATH237501.

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

University c of Leeds School of Mathematics May/June 2017 MATH237501 Linear Differential Equations and Transforms Time Allowed: 2 21 hours Answer no more than 4 questions. If you attempt 5, only the best 4 will be counted. All questions carry equal marks.

1

Turn Over

MATH237501 1. (a) For the equation S(x)y ′′ + P (x)y ′ + Q(x)y = 0,

S, P, Q polynomials,

(1)

having no common factors, what is the condition on S(x), which distinguishes between x0 being an ordinary or a singular point? If x0 is a singular point, give the conditions for it to be regular. (b) Given three (twice differentiable) functions y1 (x), y2 (x), y3 (x), define the Wronskians W [y1 , y2 ] and W [y1 , y2 , y3 ]. For the specific functions y1 = x, y2 = e−2x and a third function y(x), write the equation W [y1 , y2 , y ] = 0 as a differential equation for y(x) in the form of (1). Determine the polynomial coefficients for this case. Show that the equation has only one singular point and determine whether or not it is regular. Show that y1 (x) and y2 (x) are solutions of this differential equation. (c) Classify the point x = 0 for Legendre’s equation (1 − x2 )

d2 y dy − 2x + 6y = 0, 2 dx dx

and hence consider an analytic solution of the form y=

∞ X

an xn .

n=0

Find the general recurrence relation for the coefficients an and use this to find a2 , . . . , a5 , for arbitrary a0 , a1 . If we define y1 , y2 to be the solutions satisfying y1 (0) = 1,

y1′ (0) = 0 and y2 (0) = 0,

y2′ (0) = 1,

then show that y1 truncates to a polynomial, and write the explicit form of the series y2 up to x5 . 2. Bessel’s equation is defined by x2 y ′′ + xy ′ + (x2 − p2 )y = 0, where p is a positive constant. (a) Consider the Frobenius solution of the form y(x) = xr

∞ X

n=0

an xn ,

a0 6= 0,

and show that r = p is a solution of the indicial equation. For r = p, find the general recurrence relation and show that an = 0 when n is odd. Explicitly calculate a2 and a4 . When p = 1/2, and setting a0 = 1, show that a2 = −1/3! and a4 = 1/5!. 2

Turn Over

MATH237501 (b) When p = 1/2, define u(x) by y(x) = x−1/2 u(x) and use Bessel’s equation to show that u′′ + u = 0. Write down the general solution for u(x) and hence the general solution of Bessel’s equation (with p = 1/2) in terms of elementary functions. Which of these is the solution found in part (a)? 3. (a) Consider the general Sturm-Liouville equation !

dy d p + qy + λσy = 0, dx dx

a < x < b,

where p, q and σ are functions of x. Suppose we have a sequence of eigenfunctions yn , corresponding to distinct eigenvalues λn . Derive the formula (λn − λm )σym yn =

d ′ (p(yn ym − ym yn′ )). dx

(b) Derive the Sturm-Liouville form of Hermite’s equation y ′′ − xy ′ + λy = 0. Let Hn be a polynomial eigenfunction corresponding to eigenvalue λn = n. Deduce that Z ∞ x2 Hm Hn e− 2 dx = 0, for m 6= n. −∞

(c) Consider the generating function ∞ X

1 2

G(x, t) = ext− 2 t =

Hn (x)

n=0

tn . n!

Show that and use this to show that

Gt = (x − t)G, Hn+1 = xHn − nHn−1 .

Starting with H0 = 1, H1 = x, derive the forms of H2 , H3 , H4 and show directly that H4 satisfies Hermite’s equation for the appropriate eigenvalue. Find the value of Z ∞ 2 H22e−x /2 dx. −∞

[Hint: You may find the following formulae useful: In = (n − 1)In−2 , where In =

Z ∞

2

xn e−x

/2

dx, and

−∞

Z ∞

2

e−x

−∞

/2

dx =

√

2π.



4. Consider the following boundary value problem for Laplace’s equation: uxx + uyy = 0 for 0 < x < 2, 0 < y < 1, u(0, y ) = u(2, y ) = u(x, 0) = 0, 3

u(x, 1) = ϕ(x). Turn Over

MATH237501 (a) Use the method of separation of variables to show that the general form of the solution satisfying the three zero boundary conditions is u(x, y) =

∞ X

cn sin

n=1



 nπx nπy . sinh 2 2 

(b) Use the trigonometric identities 1 (cos(A − B) − cos(A + B )), 2 1 sin2 A = (1 − cos 2A), 2

sin A sin B =

to show the orthogonality relations Z 2 0

nπx mπx sin sin dx = δmn = 2 2 







(

0 if m 6= n, 1 if m = n,

where m and n are positive integers. (c) Find the form of cn corresponding to ϕ(x) = 1,

for 0 < x < 2,

and hence write down the solution u(x, y). 5. The Fourier Transform of a piecewise smooth function f (x), for which exists and is finite, is given by

R∞

−∞

|f (x)|dx

1 Z∞ F[f (x)] = √ f (x)e−ikx dx. 2π −∞ (a) Show that F [f (x − a)] = e−iak F [f (x)]. (b) Show that if f (x) is a differentiable function, with lim f (x) = 0,

x→±∞

then F [f ′ (x)] = ik F [f (x)]. (c) Consider the differential equation 2 ∂ 2u 2∂ u , = c ∂x2 ∂t2

subject to initial conditions u(x, 0) = f (x), ut (x, 0) = 0. Defining uˆ(k, t) = F[u(x, t)] and fˆ(k) = F[f (x)], show that 1 uˆ(k, t) = fˆ(k)(eikct + e−ikct), 2 and hence use part (a) to deduce the form of u(x, t).

4

End....